© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a264 K11a266

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X20,12,21,11 X18,14,19,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21

Gauss Code:

{1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -7, 10, -6, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 8 16 14 4 20 18 2 22 12 10

Alexander Polynomial:

- 2t-3 + 11t-2 - 25t-1 + 33 - 25t + 11t2 - 2t3

Conway Polynomial:

1 + z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a56, K11a185, ...}

Determinant and Signature:

{109, 0}

Jones Polynomial:

q-4 - 4q-3 + 8q-2 - 12q-1 + 16 - 17q + 17q2 - 14q3 + 10q4 - 6q5 + 3q6 - q7

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a185, ...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + q-8 + q-6 - 3q-4 + 4q-2 - 1 + q2 + q4 - 2q6 + 3q8 - 2q10 + q12 + 2q14 - 2q16 + q18 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 2a-4 + 4a-4z2 + 2a-4z4 - a-2 - 2a-2z2 - 2a-2z4 - a-2z6 + 1 - z2 - 2z4 - z6 + a2z2 + a2z4

Kauffman Polynomial:

- a-7z + 4a-7z3 - 4a-7z5 + a-7z7 + a-6 - 6a-6z2 + 14a-6z4 - 12a-6z6 + 3a-6z8 - 3a-5z3 + 13a-5z5 - 14a-5z7 + 4a-5z9 + 2a-4 - 14a-4z2 + 31a-4z4 - 22a-4z6 + a-4z8 + 2a-4z10 + 2a-3z - 12a-3z3 + 30a-3z5 - 31a-3z7 + 10a-3z9 + a-2 - 10a-2z2 + 27a-2z4 - 29a-2z6 + 7a-2z8 + 2a-2z10 + a-1z - 3a-1z5 - 6a-1z7 + 6a-1z9 + 1 + z4 - 11z6 + 9z8 + 3az3 - 12az5 + 10az7 + 2a2z2 - 8a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{1, 2}

Khovanov Homology: (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 11265. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   4 1   j = 9                 6 2     j = 7               8 4       j = 5             9 6         j = 3           8 8           j = 1         8 9             j = -1       5 9               j = -3     3 7                 j = -5   1 5                   j = -7   3                     j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 265]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 265]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[16, 6, 17, 5], X[14, 7, 15, 8], > X[4, 9, 5, 10], X[20, 12, 21, 11], X[18, 14, 19, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 265]]
Out[4]=	GaussCode[1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -7, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 265]]
Out[5]=	DTCode[6, 8, 16, 14, 4, 20, 18, 2, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 265]][t]
Out[6]=	2 11 25 2 3 33 - -- + -- - -- - 25 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 265]][z]
Out[7]=	2 4 6 1 + z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 56], Knot[11, Alternating, 185], > Knot[11, Alternating, 265]}
In[9]:=	{KnotDet[Knot[11, Alternating, 265]], KnotSignature[Knot[11, Alternating, 265]]}
Out[9]=	{109, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 265]][q]
Out[10]=	-4 4 8 12 2 3 4 5 6 7 16 + q - -- + -- - -- - 17 q + 17 q - 14 q + 10 q - 6 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 185], Knot[11, Alternating, 265]}
In[12]:=	A2Invariant[Knot[11, Alternating, 265]][q]
Out[12]=	-12 2 -8 -6 3 4 2 4 6 8 10 12 -1 + q - --- + q + q - -- + -- + q + q - 2 q + 3 q - 2 q + q + 10 4 2 q q q 14 16 18 22 > 2 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 265]][a, z]
Out[13]=	2 2 2 4 4 -6 2 -2 2 z 4 z 2 z 2 2 4 2 z 2 z 1 - a + -- - a - z - -- + ---- - ---- + a z - 2 z + ---- - ---- + 4 6 4 2 4 2 a a a a a a 6 2 4 6 z > a z - z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 265]][a, z]
Out[14]=	2 2 2 3 -6 2 -2 z 2 z z 6 z 14 z 10 z 2 2 4 z 1 + a + -- + a - -- + --- + - - ---- - ----- - ----- + 2 a z + ---- - 4 7 3 a 6 4 2 7 a a a a a a a 3 3 4 4 4 3 z 12 z 3 3 3 4 14 z 31 z 27 z 2 4 > ---- - ----- + 3 a z - 2 a z + z + ----- + ----- + ----- - 8 a z + 5 3 6 4 2 a a a a a 5 5 5 5 6 4 4 4 z 13 z 30 z 3 z 5 3 5 6 12 z > a z - ---- + ----- + ----- - ---- - 12 a z + 4 a z - 11 z - ----- - 7 5 3 a 6 a a a a 6 6 7 7 7 7 22 z 29 z 2 6 z 14 z 31 z 6 z 7 8 > ----- - ----- + 8 a z + -- - ----- - ----- - ---- + 10 a z + 9 z + 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 3 z z 7 z 4 z 10 z 6 z 2 z 2 z > ---- + -- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 265]], Vassiliev[3][Knot[11, Alternating, 265]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 265]][q, t]
Out[16]=	9 1 3 1 5 3 7 5 3 - + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + 8 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 8 q t + 9 q t + 6 q t + 8 q t + 4 q t + 6 q t + 2 q t + 11 5 11 6 13 6 15 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a265

K11a264 K11a266