K11a24

K11a23

K11a25

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a24 at Knotilus!

Knot presentations

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

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:K11a24/ThurstonBennequinNumber Hyperbolic Volume 17.9006 A-Polynomial See Data:K11a24/A-polynomial

[edit Notes for K11a24'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 K11a24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 157, 0 }
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+12z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+12a^{2}z^{8}+18z^{8}a^{-2}+7z^{8}a^{-4}+23z^{8}+9a^{3}z^{7}+2az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-58z^{6}+a^{5}z^{5}-13a^{3}z^{5}-26az^{5}-24z^{5}a^{-1}-21z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+42z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+9a^{3}z^{3}+25az^{3}+30z^{3}a^{-1}+22z^{3}a^{-3}+7z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$
The A2 invariant	$-q^{14}+2q^{12}-4q^{10}+2q^{8}+q^{6}-3q^{4}+7q^{2}-3+5q^{-2}-q^{-4}-2q^{-6}+3q^{-8}-5q^{-10}+2q^{-12}-q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+17q^{72}-19q^{70}+12q^{68}+11q^{66}-46q^{64}+94q^{62}-139q^{60}+154q^{58}-123q^{56}+20q^{54}+160q^{52}-369q^{50}+543q^{48}-584q^{46}+408q^{44}-30q^{42}-482q^{40}+949q^{38}-1167q^{36}+1001q^{34}-454q^{32}-314q^{30}+1007q^{28}-1337q^{26}+1155q^{24}-520q^{22}-297q^{20}+930q^{18}-1100q^{16}+723q^{14}+43q^{12}-830q^{10}+1288q^{8}-1170q^{6}+494q^{4}+492q^{2}-1389+1846q^{-2}-1655q^{-4}+877q^{-6}+226q^{-8}-1251q^{-10}+1840q^{-12}-1780q^{-14}+1119q^{-16}-121q^{-18}-823q^{-20}+1338q^{-22}-1252q^{-24}+639q^{-26}+208q^{-28}-899q^{-30}+1117q^{-32}-782q^{-34}+40q^{-36}+763q^{-38}-1275q^{-40}+1280q^{-42}-790q^{-44}+9q^{-46}+733q^{-48}-1173q^{-50}+1190q^{-52}-823q^{-54}+270q^{-56}+253q^{-58}-591q^{-60}+671q^{-62}-539q^{-64}+300q^{-66}-52q^{-68}-120q^{-70}+194q^{-72}-188q^{-74}+134q^{-76}-66q^{-78}+16q^{-80}+15q^{-82}-26q^{-84}+22q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}$

Further Quantum Invariants

K11a24 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["K11a24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 157, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+11z^{4}-4a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+12z^{2}-2a^{2}-4a^{-2}+a^{-4}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+12a^{2}z^{8}+18z^{8}a^{-2}+7z^{8}a^{-4}+23z^{8}+9a^{3}z^{7}+2az^{7}-12z^{7}a^{-1}-z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-50z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-58z^{6}+a^{5}z^{5}-13a^{3}z^{5}-26az^{5}-24z^{5}a^{-1}-21z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+42z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+50z^{4}-a^{5}z^{3}+9a^{3}z^{3}+25az^{3}+30z^{3}a^{-1}+22z^{3}a^{-3}+7z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-23z^{2}-3a^{3}z-8az-10za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+4a^{-2}+a^{-4}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a26, K11a315,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a26, K11a315,}

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["K11a24"];

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]=

{ $t^{4}-6t^{3}+18t^{2}-33t+41-33t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$, $q^{6}-4q^{5}+9q^{4}-16q^{3}+22q^{2}-25q+26-22q^{-1}+17q^{-2}-10q^{-3}+4q^{-4}-q^{-5}$ }

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]=

{K11a26, K11a315,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a26, K11a315,}

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 $4$ $-8$ $8$ $-{\frac {34}{3}}$ $-{\frac {38}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {64}{3}}$ $-8$ ${\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ ${\frac {751}{30}}$ $-{\frac {902}{15}}$ ${\frac {3302}{45}}$ $-{\frac {175}{18}}$ ${\frac {271}{30}}$

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=$0 is the signature of K11a24. 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                   6 1   5 7                 10 3     -7 5               12 6       6 3             13 10         -3 1           13 12           1 -1         10 14             4 -3       7 12               -5 -5     3 10                 7 -7   1 7                   -6 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a23 K11a25