# K11a87

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a87 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X22,19,1,20 X14,21,15,22 Gauss code 1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10 Dowker-Thistlethwaite code 4 10 12 16 18 2 20 8 6 22 14
A Braid Representative

### Three dimensional invariants

 Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a87/ThurstonBennequinNumber Hyperbolic Volume 15.1381 A-Polynomial See Data:K11a87/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $0$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial $-2 t^3+11 t^2-28 t+39-28 t^{-1} +11 t^{-2} -2 t^{-3}$ Conway polynomial $-2 z^6-z^4-2 z^2+1$ 2nd Alexander ideal (db, data sources) $\left\{2,t^2+t+1\right\}$ Determinant and Signature { 121, 0 } Jones polynomial $q^6-3 q^5+7 q^4-12 q^3+16 q^2-19 q+20-17 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5}$ HOMFLY-PT polynomial (db, data sources) $-z^6 a^{-2} -z^6+2 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+2 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +z^2-4 a^{-2} +2 a^{-4} +3$ Kauffman polynomial (db, data sources) $z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +7 a^2 z^8+12 z^8 a^{-2} +5 z^8 a^{-4} +14 z^8+7 a^3 z^7+6 a z^7-8 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +4 a^4 z^6-7 a^2 z^6-37 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -34 z^6+a^5 z^5-11 a^3 z^5-22 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -8 z^5 a^{-5} -6 a^4 z^4-a^2 z^4+41 z^4 a^{-2} +12 z^4 a^{-4} -3 z^4 a^{-6} +31 z^4-a^5 z^3+5 a^3 z^3+16 a z^3+12 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2+a^2 z^2-21 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -13 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3$ The A2 invariant Data:K11a87/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a87/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a28, K11a96,}

### Vassiliev invariants

 V2 and V3: (-2, -2)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $-16$ $32$ $\frac{116}{3}$ $\frac{76}{3}$ $128$ $\frac{800}{3}$ $\frac{224}{3}$ $80$ $-\frac{256}{3}$ $128$ $-\frac{928}{3}$ $-\frac{608}{3}$ $\frac{1769}{15}$ $\frac{628}{5}$ $-\frac{10204}{45}$ $\frac{1111}{9}$ $-\frac{1591}{15}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of K11a87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10123456χ
13           11
11          2 -2
9         51 4
7        72  -5
5       95   4
3      107    -3
1     109     1
-1    811      3
-3   59       -4
-5  38        5
-7 15         -4
-9 3          3
-111           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.