K11a178

K11a177

K11a179

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a178 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a178/ThurstonBennequinNumber Hyperbolic Volume 14.6424 A-Polynomial See Data:K11a178/A-polynomial

[edit Notes for K11a178's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a178's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, 2 }
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-12q^{6}+17q^{5}-19q^{4}+20q^{3}-17q^{2}+13q-8+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+2z^{2}a^{-2}+4z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+4z^{9}a^{-3}+8z^{9}a^{-5}+4z^{9}a^{-7}+7z^{8}a^{-2}+14z^{8}a^{-4}+13z^{8}a^{-6}+6z^{8}a^{-8}+7z^{7}a^{-1}+6z^{7}a^{-3}-6z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-2}-33z^{6}a^{-4}-39z^{6}a^{-6}-16z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-22z^{5}a^{-3}-16z^{5}a^{-5}-17z^{5}a^{-7}-11z^{5}a^{-9}-z^{4}a^{-2}+29z^{4}a^{-4}+39z^{4}a^{-6}+13z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-az^{3}+5z^{3}a^{-1}+17z^{3}a^{-3}+23z^{3}a^{-5}+19z^{3}a^{-7}+7z^{3}a^{-9}+z^{2}a^{-2}-13z^{2}a^{-4}-16z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$
The A2 invariant	Data:K11a178/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a178/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a178 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["K11a178"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 123, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-4q^{8}+7q^{7}-12q^{6}+17q^{5}-19q^{4}+20q^{3}-17q^{2}+13q-8+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+4z^{9}a^{-3}+8z^{9}a^{-5}+4z^{9}a^{-7}+7z^{8}a^{-2}+14z^{8}a^{-4}+13z^{8}a^{-6}+6z^{8}a^{-8}+7z^{7}a^{-1}+6z^{7}a^{-3}-6z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-2}-33z^{6}a^{-4}-39z^{6}a^{-6}-16z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-22z^{5}a^{-3}-16z^{5}a^{-5}-17z^{5}a^{-7}-11z^{5}a^{-9}-z^{4}a^{-2}+29z^{4}a^{-4}+39z^{4}a^{-6}+13z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-az^{3}+5z^{3}a^{-1}+17z^{3}a^{-3}+23z^{3}a^{-5}+19z^{3}a^{-7}+7z^{3}a^{-9}+z^{2}a^{-2}-13z^{2}a^{-4}-16z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}-za^{-1}-4za^{-3}-8za^{-5}-5za^{-7}+3a^{-4}+2a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a294,}

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["K11a178"];

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]=

{ $2t^{3}-11t^{2}+29t-39+29t^{-1}-11t^{-2}+2t^{-3}$, $q^{9}-4q^{8}+7q^{7}-12q^{6}+17q^{5}-19q^{4}+20q^{3}-17q^{2}+13q-8+4q^{-1}-q^{-2}$ }

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]=

{K11a294,}

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:

(3, 5)

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 $12$ $40$ $72$ $174$ $26$ $480$ ${\frac {2608}{3}}$ ${\frac {544}{3}}$ $104$ $288$ $800$ $2088$ $312$ ${\frac {42751}{10}}$ ${\frac {1774}{15}}$ ${\frac {8754}{5}}$ ${\frac {11}{2}}$ ${\frac {2431}{10}}$

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 K11a178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   4 1   3 13                 8 3     -5 11               9 4       5 9             10 8         -2 7           10 9           1 5         7 10             3 3       6 10               -4 1     3 8                 5 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a177 K11a179