K11a196

K11a195

K11a197

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a196 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a196/ThurstonBennequinNumber Hyperbolic Volume 17.2855 A-Polynomial See Data:K11a196/A-polynomial

[edit Notes for K11a196's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 2

[edit Notes for K11a196's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{7,t+1\}$
Determinant and Signature	{ 147, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-15+21q^{-1}-23q^{-2}+24q^{-3}-21q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+8a^{4}z^{2}-8a^{2}z^{2}+3z^{2}-a^{6}+2a^{4}-a^{2}+1$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+7a^{5}z^{9}+13a^{3}z^{9}+6az^{9}+10a^{6}z^{8}+18a^{4}z^{8}+15a^{2}z^{8}+7z^{8}+8a^{7}z^{7}-16a^{3}z^{7}-4az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-15a^{6}z^{6}-47a^{4}z^{6}-44a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-12a^{7}z^{5}-18a^{5}z^{5}-9a^{3}z^{5}-13az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+10a^{6}z^{4}+41a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-a^{9}z^{3}+7a^{7}z^{3}+18a^{5}z^{3}+16a^{3}z^{3}+12az^{3}+6z^{3}a^{-1}+a^{8}z^{2}-4a^{6}z^{2}-15a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-4z^{2}-2a^{7}z-4a^{5}z-4a^{3}z-3az-za^{-1}+a^{6}+2a^{4}+a^{2}+1$
The A2 invariant	$-q^{24}+q^{22}-2q^{18}+4q^{16}-4q^{14}+q^{12}+q^{10}-2q^{8}+6q^{6}-4q^{4}+4q^{2}-1-2q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-3q^{126}+7q^{124}-13q^{122}+16q^{120}-16q^{118}+7q^{116}+16q^{114}-45q^{112}+83q^{110}-114q^{108}+116q^{106}-81q^{104}-10q^{102}+143q^{100}-285q^{98}+395q^{96}-412q^{94}+288q^{92}-19q^{90}-340q^{88}+676q^{86}-847q^{84}+752q^{82}-393q^{80}-154q^{78}+682q^{76}-983q^{74}+934q^{72}-512q^{70}-90q^{68}+619q^{66}-839q^{64}+636q^{62}-113q^{60}-501q^{58}+913q^{56}-917q^{54}+476q^{52}+253q^{50}-968q^{48}+1385q^{46}-1320q^{44}+767q^{42}+61q^{40}-878q^{38}+1394q^{36}-1426q^{34}+992q^{32}-239q^{30}-517q^{28}+1002q^{26}-1043q^{24}+647q^{22}-12q^{20}-573q^{18}+842q^{16}-688q^{14}+185q^{12}+451q^{10}-920q^{8}+1023q^{6}-718q^{4}+123q^{2}+493-913q^{-2}+997q^{-4}-744q^{-6}+304q^{-8}+158q^{-10}-480q^{-12}+587q^{-14}-495q^{-16}+292q^{-18}-70q^{-20}-97q^{-22}+174q^{-24}-178q^{-26}+130q^{-28}-66q^{-30}+18q^{-32}+14q^{-34}-25q^{-36}+22q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a196 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["K11a196"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

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]=

$\{7,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 147, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-15+21q^{-1}-23q^{-2}+24q^{-3}-21q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+8a^{4}z^{2}-8a^{2}z^{2}+3z^{2}-a^{6}+2a^{4}-a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+7a^{5}z^{9}+13a^{3}z^{9}+6az^{9}+10a^{6}z^{8}+18a^{4}z^{8}+15a^{2}z^{8}+7z^{8}+8a^{7}z^{7}-16a^{3}z^{7}-4az^{7}+4z^{7}a^{-1}+4a^{8}z^{6}-15a^{6}z^{6}-47a^{4}z^{6}-44a^{2}z^{6}+z^{6}a^{-2}-15z^{6}+a^{9}z^{5}-12a^{7}z^{5}-18a^{5}z^{5}-9a^{3}z^{5}-13az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+10a^{6}z^{4}+41a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-a^{9}z^{3}+7a^{7}z^{3}+18a^{5}z^{3}+16a^{3}z^{3}+12az^{3}+6z^{3}a^{-1}+a^{8}z^{2}-4a^{6}z^{2}-15a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-4z^{2}-2a^{7}z-4a^{5}z-4a^{3}z-3az-za^{-1}+a^{6}+2a^{4}+a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a216, K11a286,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a216,}

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["K11a196"];

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]=

{ $-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-15+21q^{-1}-23q^{-2}+24q^{-3}-21q^{-4}+15q^{-5}-9q^{-6}+4q^{-7}-q^{-8}$ }

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]=

{K11a216, K11a286,}

In[6]:=

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

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

Out[6]=

{K11a216,}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-16$ $8$ ${\frac {158}{3}}$ ${\frac {34}{3}}$ $-64$ $-{\frac {448}{3}}$ ${\frac {32}{3}}$ $-48$ ${\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {136}{3}}$ ${\frac {16591}{30}}$ $-{\frac {234}{5}}$ ${\frac {10142}{45}}$ ${\frac {689}{18}}$ ${\frac {271}{30}}$

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 $s=$-2 is the signature of K11a196. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 9 3     -6 -1               12 6       6 -3             12 10         -2 -5           12 11           1 -7         9 12             3 -9       6 12               -6 -11     3 9                 6 -13   1 6                   -5 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a195 K11a197