K11a314

K11a313

K11a315

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a314 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X16,5,17,6 X18,8,19,7 X14,10,15,9 X4,12,5,11 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22
Gauss code	1, -9, 2, -6, 3, -1, 4, -10, 5, -11, 6, -2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code	6 12 16 18 14 4 20 22 2 8 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a314/ThurstonBennequinNumber Hyperbolic Volume 18.9578 A-Polynomial See Data:K11a314/A-polynomial

[edit Notes for K11a314's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a314's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+7t^{3}-21t^{2}+36t-41+36t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-z^{6}+z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 171, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+17q^{5}-24q^{4}+27q^{3}-28q^{2}+25q-18+12q^{-1}-5q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-4z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-6z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+2z^{4}-5z^{2}a^{-2}+5z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}-2a^{-2}+a^{-4}+2$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-2}+4z^{10}a^{-4}+11z^{9}a^{-1}+21z^{9}a^{-3}+10z^{9}a^{-5}+15z^{8}a^{-2}+15z^{8}a^{-4}+11z^{8}a^{-6}+11z^{8}+5az^{7}-19z^{7}a^{-1}-41z^{7}a^{-3}-9z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-51z^{6}a^{-2}-44z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-24z^{6}-8az^{5}+5z^{5}a^{-1}+21z^{5}a^{-3}-3z^{5}a^{-5}-10z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+38z^{4}a^{-2}+34z^{4}a^{-4}+6z^{4}a^{-6}-5z^{4}a^{-8}+14z^{4}+2az^{3}+8z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}-10z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}-3z^{2}-za^{-1}-4za^{-3}-4za^{-5}-za^{-7}+2a^{-2}+a^{-4}+2$
The A2 invariant	$q^{8}-3q^{6}+4q^{4}-q^{2}+1+5q^{-2}-6q^{-4}+5q^{-6}-5q^{-8}+q^{-10}+q^{-12}-4q^{-14}+5q^{-16}-2q^{-18}+q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	Data:K11a314/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a314 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["K11a314"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+7t^{3}-21t^{2}+36t-41+36t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 171, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+17q^{5}-24q^{4}+27q^{3}-28q^{2}+25q-18+12q^{-1}-5q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-2}+4z^{10}a^{-4}+11z^{9}a^{-1}+21z^{9}a^{-3}+10z^{9}a^{-5}+15z^{8}a^{-2}+15z^{8}a^{-4}+11z^{8}a^{-6}+11z^{8}+5az^{7}-19z^{7}a^{-1}-41z^{7}a^{-3}-9z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-51z^{6}a^{-2}-44z^{6}a^{-4}-14z^{6}a^{-6}+4z^{6}a^{-8}-24z^{6}-8az^{5}+5z^{5}a^{-1}+21z^{5}a^{-3}-3z^{5}a^{-5}-10z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+38z^{4}a^{-2}+34z^{4}a^{-4}+6z^{4}a^{-6}-5z^{4}a^{-8}+14z^{4}+2az^{3}+8z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}-10z^{2}a^{-2}-8z^{2}a^{-4}+z^{2}a^{-6}+2z^{2}a^{-8}-3z^{2}-za^{-1}-4za^{-3}-4za^{-5}-za^{-7}+2a^{-2}+a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a314"];

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]=

{ $-t^{4}+7t^{3}-21t^{2}+36t-41+36t^{-1}-21t^{-2}+7t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-9q^{6}+17q^{5}-24q^{4}+27q^{3}-28q^{2}+25q-18+12q^{-1}-5q^{-2}+q^{-3}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(-1, -1)

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 $-4$ $-8$ $8$ $-{\frac {14}{3}}$ $-{\frac {10}{3}}$ $32$ ${\frac {16}{3}}$ $-{\frac {128}{3}}$ $24$ $-{\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ ${\frac {40}{3}}$ $-{\frac {31}{30}}$ $-{\frac {2858}{15}}$ ${\frac {4858}{45}}$ ${\frac {703}{18}}$ ${\frac {929}{30}}$

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 K11a314. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 11 3     8 9               13 6       -7 7             14 11         3 5           14 13           -1 3         11 14             -3 1       8 15               7 -1     4 10                 -6 -3   1 8                   7 -5   4                     -4 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a313 K11a315