K11a54

K11a53

K11a55

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a54 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X10,19,11,20 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 14 16 2 18 20 22 12 10 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a54/ThurstonBennequinNumber Hyperbolic Volume 16.7029 A-Polynomial See Data:K11a54/A-polynomial

[edit Notes for K11a54'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 K11a54's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-13t^{2}+33t-43+33t^{-1}-13t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 139, -2 }
Jones polynomial	$-q^{2}+5q-10+16q^{-1}-20q^{-2}+23q^{-3}-22q^{-4}+18q^{-5}-13q^{-6}+7q^{-7}-3q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+z^{6}a^{4}+z^{4}a^{4}+z^{2}a^{4}+a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{4}+1$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+6z^{3}a^{9}-2za^{9}+5z^{8}a^{8}-12z^{6}a^{8}+10z^{4}a^{8}-5z^{2}a^{8}+a^{8}+5z^{9}a^{7}-8z^{7}a^{7}+2z^{5}a^{7}+3z^{3}a^{7}-2za^{7}+2z^{10}a^{6}+9z^{8}a^{6}-31z^{6}a^{6}+34z^{4}a^{6}-14z^{2}a^{6}+2a^{6}+12z^{9}a^{5}-21z^{7}a^{5}+11z^{5}a^{5}-z^{3}a^{5}+za^{5}+2z^{10}a^{4}+15z^{8}a^{4}-37z^{6}a^{4}+28z^{4}a^{4}-8z^{2}a^{4}+a^{4}+7z^{9}a^{3}-14z^{5}a^{3}+5z^{3}a^{3}+za^{3}+11z^{8}a^{2}-14z^{6}a^{2}+2z^{4}a^{2}-z^{2}a^{2}+10z^{7}a-14z^{5}a+3z^{3}a+5z^{6}-5z^{4}+1+z^{5}a^{-1}$
The A2 invariant	Data:K11a54/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a54/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a54 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["K11a54"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-13t^{2}+33t-43+33t^{-1}-13t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 139, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+5q-10+16q^{-1}-20q^{-2}+23q^{-3}-22q^{-4}+18q^{-5}-13q^{-6}+7q^{-7}-3q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{6}a^{10}-3z^{4}a^{10}+2z^{2}a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+6z^{3}a^{9}-2za^{9}+5z^{8}a^{8}-12z^{6}a^{8}+10z^{4}a^{8}-5z^{2}a^{8}+a^{8}+5z^{9}a^{7}-8z^{7}a^{7}+2z^{5}a^{7}+3z^{3}a^{7}-2za^{7}+2z^{10}a^{6}+9z^{8}a^{6}-31z^{6}a^{6}+34z^{4}a^{6}-14z^{2}a^{6}+2a^{6}+12z^{9}a^{5}-21z^{7}a^{5}+11z^{5}a^{5}-z^{3}a^{5}+za^{5}+2z^{10}a^{4}+15z^{8}a^{4}-37z^{6}a^{4}+28z^{4}a^{4}-8z^{2}a^{4}+a^{4}+7z^{9}a^{3}-14z^{5}a^{3}+5z^{3}a^{3}+za^{3}+11z^{8}a^{2}-14z^{6}a^{2}+2z^{4}a^{2}-z^{2}a^{2}+10z^{7}a-14z^{5}a+3z^{3}a+5z^{6}-5z^{4}+1+z^{5}a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a172,}

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["K11a54"];

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]=

{ $2t^{3}-13t^{2}+33t-43+33t^{-1}-13t^{-2}+2t^{-3}$, $-q^{2}+5q-10+16q^{-1}-20q^{-2}+23q^{-3}-22q^{-4}+18q^{-5}-13q^{-6}+7q^{-7}-3q^{-8}+q^{-9}$ }

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]=

{K11a172,}

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:

(-1, 3)

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 $-4$ $24$ $8$ $-{\frac {158}{3}}$ ${\frac {38}{3}}$ $-96$ $-80$ $-64$ $-72$ $-{\frac {32}{3}}$ $288$ ${\frac {632}{3}}$ $-{\frac {152}{3}}$ ${\frac {29969}{30}}$ $-{\frac {526}{5}}$ ${\frac {32698}{45}}$ ${\frac {1039}{18}}$ ${\frac {2129}{30}}$

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 K11a54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 10 4     6 -3               11 7       -4 -5             12 9         3 -7           10 11           1 -9         8 12             -4 -11       5 10               5 -13     2 8                 -6 -15   1 5                   4 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a53 K11a55