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

K11a266 K11a268

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 14 20 18 22 8 4 2 16

Alexander Polynomial:

- t-4 + 7t-3 - 22t-2 + 41t-1 - 49 + 41t - 22t2 + 7t3 - t4

Conway Polynomial:

1 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{191, -2}

Jones Polynomial:

- q-8 + 5q-7 - 12q-6 + 20q-5 - 27q-4 + 31q-3 - 31q-2 + 27q-1 - 19 + 12q - 5q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 3q-18 + 5q-16 - 5q-14 + 2q-12 + q-10 - 4q-8 + 6q-6 - 6q-4 + 6q-2 - 2q2 + 4q4 - 3q6 + q8

HOMFLY-PT Polynomial:

1 + z2 + 2z4 + z6 - 3a2z2 - 6a2z4 - 4a2z6 - a2z8 + 3a4z2 + 5a4z4 + 2a4z6 - a6z2 - a6z4

Kauffman Polynomial:

- a-2z4 + a-2z6 + 2a-1z3 - 8a-1z5 + 5a-1z7 + 1 - 3z2 + 14z4 - 23z6 + 11z8 + 8az5 - 21az7 + 12az9 - 9a2z2 + 42a2z4 - 54a2z6 + 15a2z8 + 5a2z10 - 2a3z3 + 26a3z5 - 51a3z7 + 26a3z9 - 9a4z2 + 42a4z4 - 60a4z6 + 21a4z8 + 5a4z10 + 4a5z3 - 5a5z5 - 13a5z7 + 14a5z9 - 3a6z2 + 12a6z4 - 25a6z6 + 17a6z8 + 4a7z3 - 14a7z5 + 12a7z7 - 3a8z4 + 5a8z6 + a9z5

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

{0, 0}

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 11267. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     4   j = 3                   8 1   j = 1                 11 4     j = -1               16 8       j = -3             16 12         j = -5           15 15           j = -7         12 16             j = -9       8 15               j = -11     4 12                 j = -13   1 8                   j = -15   4                     j = -17 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, 267]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 267]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[18, 11, 19, 12], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[4, 18, 5, 17], X[2, 19, 3, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 267]]
Out[4]=	GaussCode[1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -6, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 267]]
Out[5]=	DTCode[6, 10, 12, 14, 20, 18, 22, 8, 4, 2, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 267]][t]
Out[6]=	-4 7 22 41 2 3 4 -49 - t + -- - -- + -- + 41 t - 22 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 267]][z]
Out[7]=	6 8 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 267]}
In[9]:=	{KnotDet[Knot[11, Alternating, 267]], KnotSignature[Knot[11, Alternating, 267]]}
Out[9]=	{191, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 267]][q]
Out[10]=	-8 5 12 20 27 31 31 27 2 3 -19 - q + -- - -- + -- - -- + -- - -- + -- + 12 q - 5 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 267]}
In[12]:=	A2Invariant[Knot[11, Alternating, 267]][q]
Out[12]=	-24 2 3 5 5 2 -10 4 6 6 6 2 4 -q + --- - --- + --- - --- + --- + q - -- + -- - -- + -- - 2 q + 4 q - 22 18 16 14 12 8 6 4 2 q q q q q q q q q 6 8 > 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 267]][a, z]
Out[13]=	2 2 2 4 2 6 2 4 2 4 4 4 6 4 6 1 + z - 3 a z + 3 a z - a z + 2 z - 6 a z + 5 a z - a z + z - 2 6 4 6 2 8 > 4 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 267]][a, z]
Out[14]=	3 2 2 2 4 2 6 2 2 z 3 3 5 3 7 3 1 - 3 z - 9 a z - 9 a z - 3 a z + ---- - 2 a z + 4 a z + 4 a z + a 4 5 4 z 2 4 4 4 6 4 8 4 8 z 5 > 14 z - -- + 42 a z + 42 a z + 12 a z - 3 a z - ---- + 8 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 26 a z - 5 a z - 14 a z + a z - 23 z + -- - 54 a z - 60 a z - 2 a 7 6 6 8 6 5 z 7 3 7 5 7 7 7 > 25 a z + 5 a z + ---- - 21 a z - 51 a z - 13 a z + 12 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 11 z + 15 a z + 21 a z + 17 a z + 12 a z + 26 a z + 14 a z + 2 10 4 10 > 5 a z + 5 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 267]], Vassiliev[3][Knot[11, Alternating, 267]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 267]][q, t]
Out[16]=	12 16 1 4 1 8 4 12 8 15 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 12 16 15 15 16 8 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 11 q t + 4 q t + 8 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a267

K11a266 K11a268