K11a175

K11a174

K11a176

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a175 at Knotilus!

Knot presentations

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

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:K11a175/ThurstonBennequinNumber Hyperbolic Volume 13.9605 A-Polynomial See Data:K11a175/A-polynomial

[edit Notes for K11a175'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 K11a175's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 105, 0 }
Jones polynomial	$q^{6}-4q^{5}+7q^{4}-11q^{3}+15q^{2}-16q+17-14q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+14z^{4}-5a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+14z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+4a^{2}z^{8}+10z^{8}a^{-2}+6z^{8}a^{-4}+8z^{8}+4a^{3}z^{7}-12z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-3a^{2}z^{6}-35z^{6}a^{-2}-17z^{6}a^{-4}+z^{6}a^{-6}-23z^{6}+a^{5}z^{5}-5a^{3}z^{5}-3az^{5}+3z^{5}a^{-1}-11z^{5}a^{-3}-11z^{5}a^{-5}-6a^{4}z^{4}+39z^{4}a^{-2}+13z^{4}a^{-4}-2z^{4}a^{-6}+30z^{4}-2a^{5}z^{3}+2az^{3}+7z^{3}a^{-1}+13z^{3}a^{-3}+6z^{3}a^{-5}+3a^{4}z^{2}-3a^{2}z^{2}-17z^{2}a^{-2}-4z^{2}a^{-4}-19z^{2}+a^{5}z+a^{3}z-az-3za^{-1}-2za^{-3}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{14}+q^{12}-2q^{10}+q^{8}+q^{6}-2q^{4}+4q^{2}-2+3q^{-2}+q^{-4}+3q^{-8}-3q^{-10}-q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-2q^{68}+16q^{66}-28q^{64}+40q^{62}-44q^{60}+31q^{58}-7q^{56}-33q^{54}+73q^{52}-105q^{50}+116q^{48}-99q^{46}+48q^{44}+28q^{42}-114q^{40}+187q^{38}-217q^{36}+188q^{34}-106q^{32}-21q^{30}+149q^{28}-237q^{26}+260q^{24}-186q^{22}+57q^{20}+82q^{18}-179q^{16}+186q^{14}-106q^{12}-28q^{10}+153q^{8}-206q^{6}+154q^{4}-3q^{2}-182+332q^{-2}-366q^{-4}+266q^{-6}-62q^{-8}-177q^{-10}+371q^{-12}-440q^{-14}+374q^{-16}-184q^{-18}-46q^{-20}+244q^{-22}-332q^{-24}+289q^{-26}-145q^{-28}-33q^{-30}+168q^{-32}-213q^{-34}+144q^{-36}+5q^{-38}-159q^{-40}+256q^{-42}-247q^{-44}+122q^{-46}+54q^{-48}-226q^{-50}+318q^{-52}-304q^{-54}+192q^{-56}-27q^{-58}-131q^{-60}+228q^{-62}-241q^{-64}+183q^{-66}-85q^{-68}-13q^{-70}+77q^{-72}-103q^{-74}+90q^{-76}-55q^{-78}+24q^{-80}+5q^{-82}-17q^{-84}+17q^{-86}-14q^{-88}+7q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a175 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["K11a175"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 105, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+7q^{4}-11q^{3}+15q^{2}-16q+17-14q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+14z^{4}-5a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+14z^{2}-2a^{2}-2a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+4a^{2}z^{8}+10z^{8}a^{-2}+6z^{8}a^{-4}+8z^{8}+4a^{3}z^{7}-12z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-3a^{2}z^{6}-35z^{6}a^{-2}-17z^{6}a^{-4}+z^{6}a^{-6}-23z^{6}+a^{5}z^{5}-5a^{3}z^{5}-3az^{5}+3z^{5}a^{-1}-11z^{5}a^{-3}-11z^{5}a^{-5}-6a^{4}z^{4}+39z^{4}a^{-2}+13z^{4}a^{-4}-2z^{4}a^{-6}+30z^{4}-2a^{5}z^{3}+2az^{3}+7z^{3}a^{-1}+13z^{3}a^{-3}+6z^{3}a^{-5}+3a^{4}z^{2}-3a^{2}z^{2}-17z^{2}a^{-2}-4z^{2}a^{-4}-19z^{2}+a^{5}z+a^{3}z-az-3za^{-1}-2za^{-3}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a306,}

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["K11a175"];

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]=

{ $t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-4q^{5}+7q^{4}-11q^{3}+15q^{2}-16q+17-14q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a306,}

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, 0)

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 $8$ $0$ $32$ ${\frac {76}{3}}$ $-{\frac {28}{3}}$ $0$ $32$ $32$ $0$ ${\frac {256}{3}}$ $0$ ${\frac {608}{3}}$ $-{\frac {224}{3}}$ ${\frac {2551}{15}}$ ${\frac {676}{15}}$ $-{\frac {2996}{45}}$ $-{\frac {7}{9}}$ $-{\frac {89}{15}}$

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=$0 is the signature of K11a175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   4 1   3 7                 7 3     -4 5               8 4       4 3             8 7         -1 1           9 8           1 -1         6 9             3 -3       4 8               -4 -5     2 6                 4 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

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.

K11a174 K11a176