K11a179

K11a178

K11a180

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a179 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a179/ThurstonBennequinNumber Hyperbolic Volume 10.0674 A-Polynomial See Data:K11a179/A-polynomial

[edit Notes for K11a179's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a179's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+9t^{2}-9t+9-9t^{-1}+9t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 4 }
Jones polynomial	$-q^{9}+3q^{8}-4q^{7}+6q^{6}-8q^{5}+8q^{4}-8q^{3}+7q^{2}-5q+4-2q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+12z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-13z^{2}a^{-2}+10z^{2}a^{-4}-3z^{2}a^{-6}+4z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+5z^{9}a^{-3}+3z^{9}a^{-5}-z^{8}a^{-2}+2z^{8}a^{-4}+4z^{8}a^{-6}+z^{8}-11z^{7}a^{-1}-23z^{7}a^{-3}-8z^{7}a^{-5}+4z^{7}a^{-7}-15z^{6}a^{-2}-22z^{6}a^{-4}-9z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}+18z^{5}a^{-1}+28z^{5}a^{-3}+2z^{5}a^{-5}-4z^{5}a^{-7}+4z^{5}a^{-9}+35z^{4}a^{-2}+31z^{4}a^{-4}+3z^{4}a^{-6}-2z^{4}a^{-8}+3z^{4}a^{-10}+12z^{4}-9z^{3}a^{-1}-7z^{3}a^{-3}+z^{3}a^{-5}-5z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-23z^{2}a^{-2}-13z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}a^{-8}-2z^{2}a^{-10}-10z^{2}-za^{-3}+za^{-5}+3za^{-7}+za^{-9}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{6}+q^{4}+1+q^{-4}-q^{-8}-3q^{-12}+q^{-14}+q^{-18}+q^{-20}+q^{-24}-q^{-26}$
The G2 invariant	$q^{26}-q^{24}+4q^{22}-5q^{20}+6q^{18}-4q^{16}+11q^{12}-18q^{10}+24q^{8}-21q^{6}+10q^{4}+8q^{2}-25+39q^{-2}-35q^{-4}+23q^{-6}-2q^{-8}-17q^{-10}+27q^{-12}-29q^{-14}+18q^{-16}-4q^{-18}-9q^{-20}+15q^{-22}-10q^{-24}+12q^{-28}-18q^{-30}+17q^{-32}-11q^{-34}-6q^{-36}+18q^{-38}-32q^{-40}+34q^{-42}-25q^{-44}+8q^{-46}+9q^{-48}-26q^{-50}+31q^{-52}-29q^{-54}+15q^{-56}-2q^{-58}-10q^{-60}+14q^{-62}-10q^{-64}+5q^{-66}+3q^{-68}-2q^{-70}+2q^{-72}+4q^{-74}-6q^{-76}+10q^{-78}-6q^{-80}+3q^{-82}+3q^{-84}-3q^{-86}+6q^{-88}-7q^{-90}+7q^{-92}-7q^{-94}+9q^{-96}-8q^{-98}-3q^{-100}+6q^{-102}-11q^{-104}+16q^{-106}-16q^{-108}+11q^{-110}-2q^{-112}-7q^{-114}+13q^{-116}-18q^{-118}+14q^{-120}-10q^{-122}+5q^{-124}+2q^{-126}-8q^{-128}+12q^{-130}-9q^{-132}+8q^{-134}-3q^{-136}+q^{-140}-4q^{-142}+3q^{-144}-2q^{-146}+q^{-148}$

Further Quantum Invariants

K11a179 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["K11a179"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+9t^{2}-9t+9-9t^{-1}+9t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}-z^{4}-2z^{2}+1$

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]=

{ 57, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+12z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-13z^{2}a^{-2}+10z^{2}a^{-4}-3z^{2}a^{-6}+4z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+5z^{9}a^{-3}+3z^{9}a^{-5}-z^{8}a^{-2}+2z^{8}a^{-4}+4z^{8}a^{-6}+z^{8}-11z^{7}a^{-1}-23z^{7}a^{-3}-8z^{7}a^{-5}+4z^{7}a^{-7}-15z^{6}a^{-2}-22z^{6}a^{-4}-9z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}+18z^{5}a^{-1}+28z^{5}a^{-3}+2z^{5}a^{-5}-4z^{5}a^{-7}+4z^{5}a^{-9}+35z^{4}a^{-2}+31z^{4}a^{-4}+3z^{4}a^{-6}-2z^{4}a^{-8}+3z^{4}a^{-10}+12z^{4}-9z^{3}a^{-1}-7z^{3}a^{-3}+z^{3}a^{-5}-5z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-23z^{2}a^{-2}-13z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}a^{-8}-2z^{2}a^{-10}-10z^{2}-za^{-3}+za^{-5}+3za^{-7}+za^{-9}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11a179"];

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]=

{ $t^{4}-5t^{3}+9t^{2}-9t+9-9t^{-1}+9t^{-2}-5t^{-3}+t^{-4}$, $-q^{9}+3q^{8}-4q^{7}+6q^{6}-8q^{5}+8q^{4}-8q^{3}+7q^{2}-5q+4-2q^{-1}+q^{-2}$ }

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]=

{}

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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {512}{3}}$ ${\frac {32}{3}}$ $48$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ $-{\frac {2551}{15}}$ $-{\frac {572}{5}}$ $-{\frac {8764}{45}}$ ${\frac {391}{9}}$ $-{\frac {391}{15}}$

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=$4 is the signature of K11a179. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   2 1   -1 13                 4 2     2 11               4 2       -2 9             4 4         0 7           4 4           0 5         3 4             -1 3       3 5               2 1     1 2                 -1 -1   1 3                   2 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a178 K11a180