K11n5

K11n4

K11n6

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n5 at Knotilus!

Knot K11n5.

A graph, knot K11n5.

A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X20,16,21,15 X17,12,18,13 X22,20,1,19 X16,22,17,21
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, 8, -11, -9, 6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 10 14 2 -18 6 20 -12 22 16

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:K11n5/ThurstonBennequinNumber Hyperbolic Volume 14.1156 A-Polynomial See Data:K11n5/A-polynomial

[edit Notes for K11n5'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 K11n5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+17t-21+17t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, 2 }
Jones polynomial	$-q^{8}+4q^{7}-7q^{6}+10q^{5}-12q^{4}+12q^{3}-11q^{2}+8q-4+2q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-4}-3z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-7z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}+4z^{7}a^{-3}+9z^{7}a^{-5}+6z^{7}a^{-7}-3z^{6}a^{-2}-8z^{6}a^{-4}-z^{6}a^{-6}+4z^{6}a^{-8}+z^{5}a^{-1}-9z^{5}a^{-3}-22z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}+10z^{4}a^{-2}+4z^{4}a^{-4}-10z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}-z^{3}a^{-1}+11z^{3}a^{-3}+17z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-12z^{2}a^{-2}-2z^{2}a^{-4}+6z^{2}a^{-6}+2z^{2}a^{-8}-6z^{2}-za^{-1}-5za^{-3}-5za^{-5}-za^{-7}+4a^{-2}+2a^{-4}+3$
The A2 invariant	Data:K11n5/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n5/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n5 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["K11n5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+17t-21+17t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 71, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-7q^{6}+10q^{5}-12q^{4}+12q^{3}-11q^{2}+8q-4+2q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-4}-3z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-7z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+6z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}+4z^{7}a^{-3}+9z^{7}a^{-5}+6z^{7}a^{-7}-3z^{6}a^{-2}-8z^{6}a^{-4}-z^{6}a^{-6}+4z^{6}a^{-8}+z^{5}a^{-1}-9z^{5}a^{-3}-22z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}+10z^{4}a^{-2}+4z^{4}a^{-4}-10z^{4}a^{-6}-7z^{4}a^{-8}+3z^{4}-z^{3}a^{-1}+11z^{3}a^{-3}+17z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-12z^{2}a^{-2}-2z^{2}a^{-4}+6z^{2}a^{-6}+2z^{2}a^{-8}-6z^{2}-za^{-1}-5za^{-3}-5za^{-5}-za^{-7}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_41,}

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["K11n5"];

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]=

{ $t^{3}-7t^{2}+17t-21+17t^{-1}-7t^{-2}+t^{-3}$, $-q^{8}+4q^{7}-7q^{6}+10q^{5}-12q^{4}+12q^{3}-11q^{2}+8q-4+2q^{-1}$ }

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]=

{10_41,}

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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {608}{3}}$ ${\frac {128}{3}}$ $48$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ $-{\frac {151}{15}}$ ${\frac {308}{5}}$ $-{\frac {10204}{45}}$ ${\frac {535}{9}}$ $-{\frac {1111}{15}}$

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=$2 is the signature of K11n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n4 K11n6