# K11a189

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a189 at Knotilus!

 Knot K11a189. A graph, K11a189. A part of a link and a part of a graph.

### Knot presentations

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

### Three dimensional invariants

 Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a189/ThurstonBennequinNumber Hyperbolic Volume 17.3742 A-Polynomial See Data:K11a189/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle 4}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-6t^{3}+17t^{2}-31t+39-31t^{-1}+17t^{-2}-6t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+2z^{6}+z^{4}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 149, 0 } Jones polynomial ${\displaystyle -q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+24-24q^{-1}+21q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+a^{4}-2a^{2}+2}$ Kauffman polynomial (db, data sources) ${\displaystyle 2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+15a^{2}z^{8}+10z^{8}a^{-2}+18z^{8}+4a^{5}z^{7}-4a^{3}z^{7}-15az^{7}+z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-15a^{4}z^{6}-42a^{2}z^{6}-13z^{6}a^{-2}+4z^{6}a^{-4}-43z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-8az^{5}-17z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+10a^{4}z^{4}+34a^{2}z^{4}+6z^{4}a^{-2}-5z^{4}a^{-4}+33z^{4}+6a^{5}z^{3}+11a^{3}z^{3}+11az^{3}+13z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-13a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-12z^{2}-a^{5}z-2a^{3}z-3az-3za^{-1}-za^{-3}+a^{4}+2a^{2}+2}$ The A2 invariant ${\displaystyle q^{18}-q^{16}+2q^{12}-4q^{10}+4q^{8}-q^{6}-q^{4}+3q^{2}-5+5q^{-2}-3q^{-4}+2q^{-6}+3q^{-8}-3q^{-10}+2q^{-12}-q^{-14}}$ The G2 invariant ${\displaystyle q^{94}-3q^{92}+8q^{90}-16q^{88}+22q^{86}-25q^{84}+14q^{82}+18q^{80}-66q^{78}+128q^{76}-176q^{74}+175q^{72}-102q^{70}-63q^{68}+291q^{66}-498q^{64}+599q^{62}-499q^{60}+178q^{58}+290q^{56}-759q^{54}+1036q^{52}-973q^{50}+548q^{48}+98q^{46}-728q^{44}+1080q^{42}-1000q^{40}+535q^{38}+131q^{36}-693q^{34}+892q^{32}-650q^{30}+51q^{28}+623q^{26}-1060q^{24}+1059q^{22}-580q^{20}-201q^{18}+996q^{16}-1494q^{14}+1489q^{12}-975q^{10}+111q^{8}+783q^{6}-1390q^{4}+1499q^{2}-1079+330q^{-2}+455q^{-4}-964q^{-6}+1000q^{-8}-594q^{-10}-53q^{-12}+636q^{-14}-876q^{-16}+682q^{-18}-137q^{-20}-501q^{-22}+962q^{-24}-1052q^{-26}+755q^{-28}-204q^{-30}-393q^{-32}+806q^{-34}-921q^{-36}+752q^{-38}-387q^{-40}-5q^{-42}+304q^{-44}-454q^{-46}+439q^{-48}-320q^{-50}+157q^{-52}-7q^{-54}-89q^{-56}+127q^{-58}-122q^{-60}+89q^{-62}-47q^{-64}+15q^{-66}+8q^{-68}-17q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}}$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11a30, K11a272,}

### Vassiliev invariants

 V2 and V3: (-1, 1)
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 ${\displaystyle -4}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {14}{3}}}$ ${\displaystyle -{\frac {10}{3}}}$ ${\displaystyle -32}$ ${\displaystyle -{\frac {208}{3}}}$ ${\displaystyle -{\frac {160}{3}}}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle {\frac {56}{3}}}$ ${\displaystyle {\frac {40}{3}}}$ ${\displaystyle {\frac {3809}{30}}}$ ${\displaystyle -{\frac {1898}{15}}}$ ${\displaystyle {\frac {10618}{45}}}$ ${\displaystyle -{\frac {449}{18}}}$ ${\displaystyle {\frac {1889}{30}}}$

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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of K11a189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        103  7
3       116   -5
1      1310    3
-1     1212     0
-3    912      -3
-5   612       6
-7  39        -6
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}}$ ${\displaystyle {\mathbb {Z} }^{13}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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