K11a28

K11a27

K11a29

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a28 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,14,21,13 X22,16,1,15 X10,17,11,18 X6,19,7,20 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 12 16 2 18 20 22 10 6 14

A Braid Representative	{{{braid_table}}}
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:K11a28/ThurstonBennequinNumber Hyperbolic Volume 15.8799 A-Polynomial See Data:K11a28/A-polynomial

[edit Notes for K11a28's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+9z^{4}+3a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}+2a^{4}-4a^{2}+3$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+11az^{9}+6z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+8z^{8}a^{-2}+9z^{8}+3a^{5}z^{7}-11a^{3}z^{7}-28az^{7}-7z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-13a^{4}z^{6}-28a^{2}z^{6}-13z^{6}a^{-2}+4z^{6}a^{-4}-31z^{6}-8a^{5}z^{5}+9a^{3}z^{5}+31az^{5}+2z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+11a^{4}z^{4}+38a^{2}z^{4}+7z^{4}a^{-2}-6z^{4}a^{-4}+37z^{4}+5a^{5}z^{3}-6a^{3}z^{3}-15az^{3}+3z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-7a^{4}z^{2}-22a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-16z^{2}-a^{5}z+a^{3}z+3az+za^{-1}+2a^{4}+4a^{2}+3$
The A2 invariant	$q^{18}+2q^{12}-3q^{10}+2q^{8}-2q^{6}-2q^{4}+3q^{2}-3+5q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-2q^{92}+5q^{90}-9q^{88}+11q^{86}-12q^{84}+5q^{82}+11q^{80}-33q^{78}+61q^{76}-81q^{74}+79q^{72}-47q^{70}-21q^{68}+120q^{66}-215q^{64}+274q^{62}-251q^{60}+121q^{58}+88q^{56}-314q^{54}+476q^{52}-483q^{50}+327q^{48}-38q^{46}-283q^{44}+499q^{42}-523q^{40}+337q^{38}-25q^{36}-273q^{34}+422q^{32}-359q^{30}+115q^{28}+202q^{26}-448q^{24}+502q^{22}-341q^{20}-3q^{18}+378q^{16}-653q^{14}+716q^{12}-528q^{10}+163q^{8}+264q^{6}-606q^{4}+735q^{2}-612+288q^{-2}+107q^{-4}-413q^{-6}+520q^{-8}-381q^{-10}+100q^{-12}+206q^{-14}-387q^{-16}+362q^{-18}-157q^{-20}-144q^{-22}+397q^{-24}-489q^{-26}+401q^{-28}-161q^{-30}-121q^{-32}+341q^{-34}-440q^{-36}+397q^{-38}-250q^{-40}+59q^{-42}+109q^{-44}-215q^{-46}+243q^{-48}-203q^{-50}+132q^{-52}-44q^{-54}-30q^{-56}+72q^{-58}-88q^{-60}+73q^{-62}-46q^{-64}+21q^{-66}+2q^{-68}-13q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

K11a28 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["K11a28"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+9z^{4}+3a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}+7z^{2}+2a^{4}-4a^{2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+11az^{9}+6z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+8z^{8}a^{-2}+9z^{8}+3a^{5}z^{7}-11a^{3}z^{7}-28az^{7}-7z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-13a^{4}z^{6}-28a^{2}z^{6}-13z^{6}a^{-2}+4z^{6}a^{-4}-31z^{6}-8a^{5}z^{5}+9a^{3}z^{5}+31az^{5}+2z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+11a^{4}z^{4}+38a^{2}z^{4}+7z^{4}a^{-2}-6z^{4}a^{-4}+37z^{4}+5a^{5}z^{3}-6a^{3}z^{3}-15az^{3}+3z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-7a^{4}z^{2}-22a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-16z^{2}-a^{5}z+a^{3}z+3az+za^{-1}+2a^{4}+4a^{2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_123,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a87, K11a96,}

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["K11a28"];

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]=

{ $t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-8q^{3}+13q^{2}-17q+20-19q^{-1}+16q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ }

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_123,}

In[6]:=

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

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

Out[6]=

{K11a87, K11a96,}

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 $-8$ $16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $-128$ $-{\frac {704}{3}}$ $-{\frac {320}{3}}$ $-16$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1289}{15}}$ ${\frac {388}{5}}$ $-{\frac {5884}{45}}$ ${\frac {391}{9}}$ $-{\frac {871}{15}}$

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=$0 is the signature of K11a28. 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 11                       1 -1 9                     3   3 7                   5 1   -4 5                 8 3     5 3               9 5       -4 1             11 8         3 -1           9 10           1 -3         7 10             -3 -5       5 9               4 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a27 K11a29