K11a189

K11a188

K11a190

(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

A Braid Representative
A Morse Link Presentation

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

[edit Notes for K11a189's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a189's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

K11a189 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a189"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-6t^{3}+17t^{2}-31t+39-31t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}+z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 149, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+24-24q^{-1}+21q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+a^{4}-2a^{2}+2$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+15a^{2}z^{8}+10z^{8}a^{-2}+18z^{8}+4a^{5}z^{7}-4a^{3}z^{7}-15az^{7}+z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-15a^{4}z^{6}-42a^{2}z^{6}-13z^{6}a^{-2}+4z^{6}a^{-4}-43z^{6}-9a^{5}z^{5}-12a^{3}z^{5}-8az^{5}-17z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+10a^{4}z^{4}+34a^{2}z^{4}+6z^{4}a^{-2}-5z^{4}a^{-4}+33z^{4}+6a^{5}z^{3}+11a^{3}z^{3}+11az^{3}+13z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-13a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-12z^{2}-a^{5}z-2a^{3}z-3az-3za^{-1}-za^{-3}+a^{4}+2a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a189"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $t^{4}-6t^{3}+17t^{2}-31t+39-31t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+24-24q^{-1}+21q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{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 $-4$ $8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $-32$ $-{\frac {208}{3}}$ $-{\frac {160}{3}}$ $8$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ ${\frac {3809}{30}}$ $-{\frac {1898}{15}}$ ${\frac {10618}{45}}$ $-{\frac {449}{18}}$ ${\frac {1889}{30}}$

V_2,1 through V_6,9 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 $t^{r}q^{j}$ 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 K11a189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 6       -5 1             13 10         3 -1           12 12           0 -3         9 12             -3 -5       6 12               6 -7     3 9                 -6 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a188 K11a190