K11a14

K11a13

K11a15

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a14 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,11,19,12 X6,13,7,14 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -9, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 12 2 14 18 6 20 10 22 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a14/ThurstonBennequinNumber Hyperbolic Volume 16.4066 A-Polynomial See Data:K11a14/A-polynomial

[edit Notes for K11a14's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+15t^{2}-28t+35-28t^{-1}+15t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+5z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 133, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-14q^{3}+19q^{2}-21q+22-18q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-12z^{2}a^{-2}+2z^{2}a^{-4}+20z^{2}-4a^{2}-6a^{-2}+a^{-4}+10$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+5az^{9}+9z^{9}a^{-1}+4z^{9}a^{-3}+8a^{2}z^{8}+17z^{8}a^{-2}+6z^{8}a^{-4}+19z^{8}+6a^{3}z^{7}+4az^{7}-z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-14a^{2}z^{6}-46z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-50z^{6}+a^{5}z^{5}-9a^{3}z^{5}-24az^{5}-36z^{5}a^{-1}-32z^{5}a^{-3}-10z^{5}a^{-5}-4a^{4}z^{4}+17a^{2}z^{4}+41z^{4}a^{-2}+6z^{4}a^{-4}-2z^{4}a^{-6}+54z^{4}-2a^{5}z^{3}+7a^{3}z^{3}+29az^{3}+43z^{3}a^{-1}+31z^{3}a^{-3}+8z^{3}a^{-5}+a^{4}z^{2}-13a^{2}z^{2}-21z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-32z^{2}+a^{5}z-3a^{3}z-12az-16za^{-1}-10za^{-3}-2za^{-5}+4a^{2}+6a^{-2}+a^{-4}+10$
The A2 invariant	$-q^{14}+q^{12}-4q^{10}+q^{8}+q^{6}-2q^{4}+7q^{2}-1+5q^{-2}-2q^{-6}+2q^{-8}-5q^{-10}+q^{-12}-q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+10q^{72}-10q^{70}+4q^{68}+10q^{66}-29q^{64}+52q^{62}-74q^{60}+78q^{58}-61q^{56}+7q^{54}+89q^{52}-198q^{50}+290q^{48}-316q^{46}+225q^{44}-31q^{42}-245q^{40}+508q^{38}-649q^{36}+586q^{34}-312q^{32}-108q^{30}+511q^{28}-748q^{26}+712q^{24}-413q^{22}-30q^{20}+427q^{18}-607q^{16}+494q^{14}-126q^{12}-308q^{10}+628q^{8}-659q^{6}+373q^{4}+140q^{2}-660+1001q^{-2}-986q^{-4}+624q^{-6}-15q^{-8}-611q^{-10}+1037q^{-12}-1105q^{-14}+803q^{-16}-249q^{-18}-340q^{-20}+734q^{-22}-798q^{-24}+534q^{-26}-76q^{-28}-370q^{-30}+589q^{-32}-511q^{-34}+163q^{-36}+282q^{-38}-631q^{-40}+730q^{-42}-540q^{-44}+132q^{-46}+313q^{-48}-642q^{-50}+740q^{-52}-591q^{-54}+284q^{-56}+60q^{-58}-328q^{-60}+447q^{-62}-411q^{-64}+273q^{-66}-95q^{-68}-52q^{-70}+134q^{-72}-153q^{-74}+123q^{-76}-69q^{-78}+24q^{-80}+9q^{-82}-23q^{-84}+21q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a14 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["K11a14"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+15t^{2}-28t+35-28t^{-1}+15t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+5z^{4}+3z^{2}+1$

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]=

{ 133, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+8q^{4}-14q^{3}+19q^{2}-21q+22-18q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-12z^{2}a^{-2}+2z^{2}a^{-4}+20z^{2}-4a^{2}-6a^{-2}+a^{-4}+10$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+5az^{9}+9z^{9}a^{-1}+4z^{9}a^{-3}+8a^{2}z^{8}+17z^{8}a^{-2}+6z^{8}a^{-4}+19z^{8}+6a^{3}z^{7}+4az^{7}-z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-14a^{2}z^{6}-46z^{6}a^{-2}-12z^{6}a^{-4}+z^{6}a^{-6}-50z^{6}+a^{5}z^{5}-9a^{3}z^{5}-24az^{5}-36z^{5}a^{-1}-32z^{5}a^{-3}-10z^{5}a^{-5}-4a^{4}z^{4}+17a^{2}z^{4}+41z^{4}a^{-2}+6z^{4}a^{-4}-2z^{4}a^{-6}+54z^{4}-2a^{5}z^{3}+7a^{3}z^{3}+29az^{3}+43z^{3}a^{-1}+31z^{3}a^{-3}+8z^{3}a^{-5}+a^{4}z^{2}-13a^{2}z^{2}-21z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}-32z^{2}+a^{5}z-3a^{3}z-12az-16za^{-1}-10za^{-3}-2za^{-5}+4a^{2}+6a^{-2}+a^{-4}+10$

"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["K11a14"];

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]=

{ $t^{4}-5t^{3}+15t^{2}-28t+35-28t^{-1}+15t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-4q^{5}+8q^{4}-14q^{3}+19q^{2}-21q+22-18q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$ }

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:

(3, -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 $12$ $-8$ $72$ $62$ $-6$ $-96$ $-{\frac {368}{3}}$ $-{\frac {32}{3}}$ $-8$ $288$ $32$ $744$ $-72$ ${\frac {7311}{10}}$ ${\frac {2974}{15}}$ $-{\frac {698}{15}}$ ${\frac {145}{6}}$ $-{\frac {689}{10}}$

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 K11a14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a13 K11a15