K11a15

K11a14

K11a16

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a15 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a15/ThurstonBennequinNumber Hyperbolic Volume 15.1288 A-Polynomial See Data:K11a15/A-polynomial

[edit Notes for K11a15'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 K11a15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 107, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+6a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+14z^{2}+3a^{4}-8a^{2}-2a^{-2}+8$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+5a^{3}z^{9}+8az^{9}+3z^{9}a^{-1}+9a^{4}z^{8}+15a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{5}z^{7}+a^{3}z^{7}-15az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-16a^{4}z^{6}-52a^{2}z^{6}-12z^{6}a^{-2}-42z^{6}+3a^{7}z^{5}-15a^{5}z^{5}-25a^{3}z^{5}-9az^{5}-6z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+13a^{4}z^{4}+58a^{2}z^{4}+16z^{4}a^{-2}+54z^{4}-2a^{7}z^{3}+14a^{5}z^{3}+29a^{3}z^{3}+23az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-7a^{4}z^{2}-33a^{2}z^{2}-9z^{2}a^{-2}-31z^{2}-5a^{5}z-11a^{3}z-10az-6za^{-1}-2za^{-3}+3a^{4}+8a^{2}+2a^{-2}+8$
The A2 invariant	$q^{20}-q^{18}+3q^{16}-q^{14}-q^{12}+q^{10}-5q^{8}+2q^{6}-3q^{4}+2q^{2}+3+4q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+6q^{106}-5q^{104}+9q^{100}-19q^{98}+29q^{96}-37q^{94}+33q^{92}-20q^{90}-5q^{88}+44q^{86}-78q^{84}+105q^{82}-114q^{80}+87q^{78}-31q^{76}-56q^{74}+155q^{72}-222q^{70}+240q^{68}-180q^{66}+52q^{64}+114q^{62}-253q^{60}+318q^{58}-267q^{56}+116q^{54}+78q^{52}-231q^{50}+278q^{48}-189q^{46}+10q^{44}+180q^{42}-295q^{40}+259q^{38}-97q^{36}-144q^{34}+352q^{32}-446q^{30}+366q^{28}-152q^{26}-136q^{24}+378q^{22}-501q^{20}+449q^{18}-254q^{16}-18q^{14}+262q^{12}-389q^{10}+367q^{8}-197q^{6}-28q^{4}+220q^{2}-291+223q^{-2}-32q^{-4}-173q^{-6}+318q^{-8}-320q^{-10}+188q^{-12}+29q^{-14}-235q^{-16}+359q^{-18}-346q^{-20}+222q^{-22}-39q^{-24}-136q^{-26}+236q^{-28}-246q^{-30}+181q^{-32}-80q^{-34}-17q^{-36}+76q^{-38}-97q^{-40}+81q^{-42}-48q^{-44}+17q^{-46}+5q^{-48}-16q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a15 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["K11a15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 107, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+6a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+14z^{2}+3a^{4}-8a^{2}-2a^{-2}+8$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+5a^{3}z^{9}+8az^{9}+3z^{9}a^{-1}+9a^{4}z^{8}+15a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{5}z^{7}+a^{3}z^{7}-15az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-16a^{4}z^{6}-52a^{2}z^{6}-12z^{6}a^{-2}-42z^{6}+3a^{7}z^{5}-15a^{5}z^{5}-25a^{3}z^{5}-9az^{5}-6z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+13a^{4}z^{4}+58a^{2}z^{4}+16z^{4}a^{-2}+54z^{4}-2a^{7}z^{3}+14a^{5}z^{3}+29a^{3}z^{3}+23az^{3}+15z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}-7a^{4}z^{2}-33a^{2}z^{2}-9z^{2}a^{-2}-31z^{2}-5a^{5}z-11a^{3}z-10az-6za^{-1}-2za^{-3}+3a^{4}+8a^{2}+2a^{-2}+8$

"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["K11a15"];

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]=

{ $-t^{4}+5t^{3}-13t^{2}+22t-25+22t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+3q^{3}-6q^{2}+11q-14+17q^{-1}-17q^{-2}+15q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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, 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 $-4$ $24$ $8$ ${\frac {34}{3}}$ ${\frac {86}{3}}$ $-96$ $-272$ $-96$ $-104$ $-{\frac {32}{3}}$ $288$ $-{\frac {136}{3}}$ $-{\frac {344}{3}}$ ${\frac {22289}{30}}$ ${\frac {3622}{15}}$ ${\frac {778}{45}}$ ${\frac {2383}{18}}$ $-{\frac {2671}{30}}$

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 K11a15. 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 9                       1 -1 7                     2   2 5                   4 1   -3 3                 7 2     5 1               7 4       -3 -1             10 7         3 -3           8 8           0 -5         7 9             -2 -7       5 8               3 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a14 K11a16