K11a5

K11a4

K11a6

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a5 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a5/ThurstonBennequinNumber Hyperbolic Volume 15.5012 A-Polynomial See Data:K11a5/A-polynomial

[edit Notes for K11a5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for K11a5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+9t^{2}-30t+45-30t^{-1}+9t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+3z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 125, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{6}-2z^{4}-5z^{2}-3+2z^{4}a^{-2}+2z^{2}a^{-2}+2a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+10a^{2}z^{8}+7z^{8}a^{-2}+13z^{8}+3a^{5}z^{7}+3a^{3}z^{7}+az^{7}+8z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-6a^{4}z^{6}-19a^{2}z^{6}-4z^{6}a^{-2}+4z^{6}a^{-4}-20z^{6}-8a^{5}z^{5}-18a^{3}z^{5}-23az^{5}-24z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}-2a^{4}z^{4}+3a^{2}z^{4}-8z^{4}a^{-2}-6z^{4}a^{-4}+7a^{5}z^{3}+15a^{3}z^{3}+18az^{3}+16z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+6a^{4}z^{2}+7a^{2}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+9z^{2}-2a^{5}z-3a^{3}z-4az-4za^{-1}-za^{-3}-a^{6}-2a^{4}-3a^{2}-2a^{-2}-3$
The A2 invariant	Data:K11a5/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a5/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a5 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["K11a5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+9t^{2}-30t+45-30t^{-1}+9t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 125, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{6}-2z^{4}-5z^{2}-3+2z^{4}a^{-2}+2z^{2}a^{-2}+2a^{-2}-z^{2}a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+7az^{9}+4z^{9}a^{-1}+4a^{4}z^{8}+10a^{2}z^{8}+7z^{8}a^{-2}+13z^{8}+3a^{5}z^{7}+3a^{3}z^{7}+az^{7}+8z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-6a^{4}z^{6}-19a^{2}z^{6}-4z^{6}a^{-2}+4z^{6}a^{-4}-20z^{6}-8a^{5}z^{5}-18a^{3}z^{5}-23az^{5}-24z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}-2a^{4}z^{4}+3a^{2}z^{4}-8z^{4}a^{-2}-6z^{4}a^{-4}+7a^{5}z^{3}+15a^{3}z^{3}+18az^{3}+16z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+6a^{4}z^{2}+7a^{2}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+9z^{2}-2a^{5}z-3a^{3}z-4az-4za^{-1}-za^{-3}-a^{6}-2a^{4}-3a^{2}-2a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a112,}

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["K11a5"];

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]=

{ $-t^{3}+9t^{2}-30t+45-30t^{-1}+9t^{-2}-t^{-3}$, $-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ }

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]=

{K11a112,}

Vassiliev invariants

V2 and V3:

(-3, 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 $-12$ $16$ $72$ $50$ $14$ $-192$ $-{\frac {704}{3}}$ $-{\frac {224}{3}}$ $-16$ $-288$ $128$ $-600$ $-168$ $-{\frac {1471}{10}}$ $-{\frac {454}{15}}$ $-{\frac {1022}{15}}$ ${\frac {127}{6}}$ $-{\frac {191}{10}}$

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=$0 is the signature of K11a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a4 K11a6