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

K11a267 K11a269

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X8,20,9,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 18 20 16 22 4 2 8 14

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 29t-1 - 33 + 29t - 17t2 + 6t3 - t4

Conway Polynomial:

1 - z2 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{139, 2}

Jones Polynomial:

- q-4 + 4q-3 - 8q-2 + 14q-1 - 19 + 22q - 22q2 + 20q3 - 15q4 + 9q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 - q-6 + 4q-4 - 3q-2 + 1 + q2 - 3q4 + 5q6 - 4q8 + 3q10 - q12 - 2q14 + 3q16 - 2q18 + q20

HOMFLY-PT Polynomial:

a-4 + 3a-4z2 + 3a-4z4 + a-4z6 - 2a-2 - 9a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 2 + 7z2 + 7z4 + 2z6 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 9a-6z6 - a-5z + 9a-5z3 - 19a-5z5 + 14a-5z7 + a-4 - 4a-4z2 + 13a-4z4 - 25a-4z6 + 15a-4z8 - a-3z + 9a-3z3 - 11a-3z5 - 10a-3z7 + 10a-3z9 + 2a-2 - 18a-2z2 + 51a-2z4 - 57a-2z6 + 15a-2z8 + 3a-2z10 - 7a-1z3 + 32a-1z5 - 44a-1z7 + 16a-1z9 + 2 - 17z2 + 45z4 - 37z6 + 4z8 + 3z10 - 4az3 + 17az5 - 19az7 + 6az9 - 5a2z2 + 15a2z4 - 14a2z6 + 4a2z8 + 2a3z3 - 3a3z5 + a3z7

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

{-1, -1}

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=2 is the signature of 11268. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   6 1   j = 9                 9 3     j = 7               11 6       j = 5             11 9         j = 3           11 11           j = 1         9 12             j = -1       5 10               j = -3     3 9                 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, 268]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 268]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[16, 11, 17, 12], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 268]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, 7, -11, 8, -6, 9, -4, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 268]]
Out[5]=	DTCode[6, 10, 12, 18, 20, 16, 22, 4, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 268]][t]
Out[6]=	-4 6 17 29 2 3 4 -33 - t + -- - -- + -- + 29 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 268]][z]
Out[7]=	2 4 6 8 1 - z - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 268]}
In[9]:=	{KnotDet[Knot[11, Alternating, 268]], KnotSignature[Knot[11, Alternating, 268]]}
Out[9]=	{139, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 268]][q]
Out[10]=	-4 4 8 14 2 3 4 5 6 7 -19 - q + -- - -- + -- + 22 q - 22 q + 20 q - 15 q + 9 q - 4 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, 268]}
In[12]:=	A2Invariant[Knot[11, Alternating, 268]][q]
Out[12]=	-12 -10 -8 -6 4 3 2 4 6 8 10 12 1 - q + q + q - q + -- - -- + q - 3 q + 5 q - 4 q + 3 q - q - 4 2 q q 14 16 18 20 > 2 q + 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 268]][a, z]
Out[13]=	2 2 4 4 -4 2 2 3 z 9 z 2 2 4 3 z 10 z 2 4 2 + a - -- + 7 z + ---- - ---- - 2 a z + 7 z + ---- - ----- - a z + 2 4 2 4 2 a a a a a 6 6 8 6 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 268]][a, z]
Out[14]=	2 2 2 3 3 -4 2 z z 2 2 z 4 z 18 z 2 2 z 9 z 2 + a + -- - -- - -- - 17 z + ---- - ---- - ----- - 5 a z - -- + ---- + 2 5 3 6 4 2 7 5 a a a a a a a a 3 3 4 4 4 4 9 z 7 z 3 3 3 4 z 7 z 13 z 51 z > ---- - ---- - 4 a z + 2 a z + 45 z + -- - ---- + ----- + ----- + 3 a 8 6 4 2 a a a a a 5 5 5 5 2 4 4 z 19 z 11 z 32 z 5 3 5 6 > 15 a z + ---- - ----- - ----- + ----- + 17 a z - 3 a z - 37 z + 7 5 3 a a a a 6 6 6 7 7 7 9 z 25 z 57 z 2 6 14 z 10 z 44 z 7 3 7 > ---- - ----- - ----- - 14 a z + ----- - ----- - ----- - 19 a z + a z + 6 4 2 5 3 a a a a a a 8 8 9 9 10 8 15 z 15 z 2 8 10 z 16 z 9 10 3 z > 4 z + ----- + ----- + 4 a z + ----- + ----- + 6 a z + 3 z + ----- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 268]], Vassiliev[3][Knot[11, Alternating, 268]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, Alternating, 268]][q, t]
Out[16]=	3 1 3 1 5 3 9 5 10 12 q + 11 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 9 q 3 5 5 2 7 2 7 3 9 3 > --- + 11 q t + 11 q t + 9 q t + 11 q t + 6 q t + 9 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a268

K11a267 K11a269