Property of the interior polynomial from the HOMFLY polynomial - PowerPoint PPT Presentation

Computing the HOMFLY polynomial using the combinatorics Properties of the interior polynomial Property of the interior polynomial from the HOMFLY polynomial 嘉藤桂樹 東京工業大学理学院数学系博士課程後期１年 2017 年 12 月 24 日 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Properties of the interior polynomial Computing the HOMFLY polynomial using the combinatorics 1 HOMFLY polynomial Interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Properties of the interior polynomial 2 Mirror image Flyping and mutation 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Definition 1 (HOMFLY polynomial) There is a function P : { oriented links in S 3 } → Z [ v ± 1 , z ± 1 ] defined uniquely by ( i ) P ( unknot ) = 1, ( ii ) v − 1 P D + − vP D − = zP D 0 , where D + , D − , D 0 are an oriented skein triple. D + D − D 0 Definition 2 (top of the HOMFLY polynomial) Top D ( v ) = the coefficient of z c ( D ) − s ( D )+1 in the HOMFLY polynomial of D . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Interior polynomial H = ( V , E ) : hypergraph. I H ( x ) : interior polynomial (T. K´ alm´ an 2013). It generalizes the evaluation x | V |− 1 T G (1 / x , 1) of the classical Tutte polynomial T G ( x , y ) of the graph G = ( V , E ). We regard the interior polynomial as an invariant of bipartite graph G = ( V , E , E ) with color classes E and V (T. K´ alm´ an and A. Postnikov 2016). Example I G ( x ) = 1 x 0 + 3 x 1 + 3 x 2 . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial For any plane bipartite graph G , Let L G be the alternating link obtained from G by replacing each edge by a crossing. Obviously, L G is a special alternating diagram. Theorem 3 (T. K´ alm´ an, H. Murakami and A. Postnikov, 2016) G = ( V , E , E ) : a connected plane bipartite graph. Top L G ( v ) = v |E|− ( | V | + | E | )+1 I G ( v 2 ) , where I G ( x ) is the interior polynomial of G. 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial G = ( V , E , E ) : a bipartite graph Definition 4 For v ∈ V and e ∈ E , let v and e denote the standard generators of R V ⊕ R E . Then the root polytope of G is defined to be Q G = Conv { v + e | ve is an edge of G } . Example d = dim Q G = | V | + | E | − 2. 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Q G : the root polytope of a bipartite graph G . Definition 5 (Ehrhart polynoial) ε Q G ( s ) := | s · Q G ∩ Z V ⊕ Z E | . Definition 6 (Ehrhart series) ∑ ε Q G ( s ) x s . Ehr Q G ( x ) = s ∈ Z ≥ 0 Theorem 7 (T. K´ alm´ an and A. Postnikov, 2016) G = ( V , E , E ) : connected bipartite graph. I G : the interior polynomial of G. I G ( x ) (1 − x ) d +1 = Ehr Q G ( x ) . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial G = ( V , E , E − ∪ E + ) : a connected signed bipartite graph. Definition 8 (signed interior polynomial) I + ∑ ( − 1) |S| I G \S ( x ) , G ( x ) = S⊆E − where G \ S is bipartite graph obtained from G by deleting ∀ e ∈ S and by forgetting sign. Example I + G = 1 x 3 . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial For any signed plane bipartite graph G , Let L G be the oriented link obtained from G by replacing each edge to a crossing. positive edge negative edge Obviously, L G is a special diagram. Theorem 9 (K.) G = ( V , E , E + ∪ E − ) : plane signed bipartite graph. Top L G ( v ) = v |E + |−|E − |− ( | V | + | E | )+1 I + v 2 ) ( , G where I + G ( x ) is the interior polynomial of G. 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Example I + G = 1 x 3 . +1 v 3 z 3 P L G ( v , z ) = +4 v 3 z − 1 v 5 z − 1 vz − 1 +3 v 3 z − 1 − 2 v 5 z − 1 . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Proposition 10 (Murasugi and Pryzytycki, 1989) D 1 ∗ D 2 : a link diagram obtained by Murasugi-sum. Then Top D 1 ∗ D 2 ( v ) = Top D 1 ( v ) Top D 2 ( v ) . Proposition 11 G 1 ∗ G 2 : a signed bipartite graph obtained by identifying one vertex. Then I + G 1 ∗ G 2 ( x ) = I + G 1 ( x ) I + G 2 ( x ) . 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial Theorem 12 (K.) D : oriented link diagram. G = ( V , E , E + ∪ E − ) : the Seifert graph of D. Then Top D ( v ) = v |E + |−|E − |− ( | V | + | E | )+1 I + v 2 ) ( . G 嘉藤桂樹 Property of the interior polynomial

HOMFLY polynomial Computing the HOMFLY polynomial using the combinatorics Interior polynomial Properties of the interior polynomial Root polytope and Ehrhart polynomial Signed version of interior polynomial and Ehrhart polynomial G = ( V , E , E + ∪ E − ) : a signed bipartite graph. Definition 13 (the signed Ehrhart series) ( − 1) |S| Ehr Q G \S ( x ) . ∑ Ehr + G ( x ) = S⊆E − ( G ) Theorem 14 (K.) I + G ( x ) : the signed interior polynomial of G. Then I + G ( x ) (1 − x ) d +1 = Ehr + G ( x ) . 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Mirror image Properties of the interior polynomial Flyping and mutation Theorem 15 L ∗ : mirror image of L. Then P L ∗ ( v , z ) = P L ( − v − 1 , z ) . Example L ∗ L 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Mirror image Properties of the interior polynomial Flyping and mutation Theorem 16 (Ehrhart reciprocity) P : rational convex polytope Ehr P (1 / x ) = ( − 1) dim P +1 Ehr int P ( x ) . G = ( V , E , E = E + ) : bipartite graph with only positive edge. Q G : the root polytope of G (forgetting sign). Ehr Q G (1 / x ) = ( − 1) d +1 Ehr int Q G ( x ) . Lemma 17 ( − 1) d Ehr int Q G ( x ) = ( − 1) |S|− 1 Ehr Q S ( x ) , ∑ S⊂E where Q S is the root polytope of the bipartite graph whose edges consist of S . 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Mirror image Properties of the interior polynomial Flyping and mutation Therefore, ( − 1) |S| Ehr Q S ( x ) . ∑ Ehr Q G (1 / x ) = S⊂E By definition of the signed Ehrhart series, ( − 1) |E| Ehr Q G (1 / x ) ( − 1) |E|−|S| Ehr Q S ( x ) ∑ = S⊂E Ehr + = Q − G ( x ) , where Q − G is the root polytope of the bipartite graph obtained from G by changing sign. 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Mirror image Properties of the interior polynomial Flyping and mutation By using Theorem 14, I + I + − G ( x ) G (1 / x ) ( − 1) |E| (1 − 1 / x ) d +1 = (1 − x ) d +1 . We get ( − 1) |E| + d +1 x d +1 I + G (1 / x ) = I + − G ( x ) . And by using induction on |E − | , we prove the following theorem. Theorem 18 (K.) G = ( V , E , E + ∪ E − ) : signed bipartite graph. − G : the signed bipartite graph obtained from G by changing sign. Then ( − 1) |E + | + |E − | + | E | + | V |− 1 x | E | + | V |− 1 I + G (1 / x ) = I + − G ( x ) . 嘉藤桂樹 Property of the interior polynomial

Computing the HOMFLY polynomial using the combinatorics Mirror image Properties of the interior polynomial Flyping and mutation Flyping ← → isotopy ← → 嘉藤桂樹 Property of the interior polynomial