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

K11a265 K11a267

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 12 14 16 18 22 20 4 2 8

Alexander Polynomial:

t-4 - 7t-3 + 23t-2 - 45t-1 + 57 - 45t + 23t2 - 7t3 + t4

Conway Polynomial:

1 + z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{209, 0}

Jones Polynomial:

q-6 - 6q-5 + 14q-4 - 22q-3 + 30q-2 - 34q-1 + 34 - 29q + 21q2 - 12q3 + 5q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 - 3q-16 + 3q-12 - 5q-10 + 7q-8 - q-6 + 4q-2 - 7 + 6q2 - 6q4 + 2q6 + 4q8 - 4q10 + 3q12 - q14

HOMFLY-PT Polynomial:

a-2 - a-2z2 - 2a-2z4 - a-2z6 - 2 + z2 + 6z4 + 4z6 + z8 + 3a2 - 4a2z4 - 2a2z6 - a4 + a4z4

Kauffman Polynomial:

a-5z5 - 3a-4z4 + 5a-4z6 + 4a-3z3 - 13a-3z5 + 12a-3z7 - a-2 - 2a-2z2 + 14a-2z4 - 26a-2z6 + 18a-2z8 - a-1z + 7a-1z3 - 4a-1z5 - 16a-1z7 + 16a-1z9 - 2 - 3z2 + 44z4 - 69z6 + 24z8 + 6z10 - az + 5az3 + 19az5 - 57az7 + 31az9 - 3a2 + a2z2 + 37a2z4 - 65a2z6 + 20a2z8 + 6a2z10 + a3z + 2a3z3 + 2a3z5 - 23a3z7 + 15a3z9 - a4 + 2a4z2 + 10a4z4 - 26a4z6 + 14a4z8 + a5z - 7a5z5 + 6a5z7 + a6z6

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

{0, -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=0 is the signature of 11266. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     4   j = 7                   8 1   j = 5                 13 4     j = 3               16 8       j = 1             18 13         j = -1           17 17           j = -3         13 17             j = -5       9 17               j = -7     5 13                 j = -9   1 9                   j = -11   5                     j = -13 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, 266]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 266]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[4, 18, 5, 17], X[2, 19, 3, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 266]]
Out[4]=	GaussCode[1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 266]]
Out[5]=	DTCode[6, 10, 12, 14, 16, 18, 22, 20, 4, 2, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 266]][t]
Out[6]=	-4 7 23 45 2 3 4 57 + t - -- + -- - -- - 45 t + 23 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 266]][z]
Out[7]=	4 6 8 1 + z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 266]}
In[9]:=	{KnotDet[Knot[11, Alternating, 266]], KnotSignature[Knot[11, Alternating, 266]]}
Out[9]=	{209, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 266]][q]
Out[10]=	-6 6 14 22 30 34 2 3 4 5 34 + q - -- + -- - -- + -- - -- - 29 q + 21 q - 12 q + 5 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 266]}
In[12]:=	A2Invariant[Knot[11, Alternating, 266]][q]
Out[12]=	-18 3 3 5 7 -6 4 2 4 6 8 -7 + q - --- + --- - --- + -- - q + -- + 6 q - 6 q + 2 q + 4 q - 16 12 10 8 2 q q q q q 10 12 14 > 4 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 266]][a, z]
Out[13]=	2 4 6 -2 2 4 2 z 4 2 z 2 4 4 4 6 z -2 + a + 3 a - a + z - -- + 6 z - ---- - 4 a z + a z + 4 z - -- - 2 2 2 a a a 2 6 8 > 2 a z + z
In[14]:=	Kauffman[Knot[11, Alternating, 266]][a, z]
Out[14]=	2 -2 2 4 z 3 5 2 2 z 2 2 4 2 -2 - a - 3 a - a - - - a z + a z + a z - 3 z - ---- + a z + 2 a z + a 2 a 3 3 4 4 4 z 7 z 3 3 3 4 3 z 14 z 2 4 > ---- + ---- + 5 a z + 2 a z + 44 z - ---- + ----- + 37 a z + 3 a 4 2 a a a 5 5 5 6 4 4 z 13 z 4 z 5 3 5 5 5 6 5 z > 10 a z + -- - ----- - ---- + 19 a z + 2 a z - 7 a z - 69 z + ---- - 5 3 a 4 a a a 6 7 7 26 z 2 6 4 6 6 6 12 z 16 z 7 3 7 > ----- - 65 a z - 26 a z + a z + ----- - ----- - 57 a z - 23 a z + 2 3 a a a 8 9 5 7 8 18 z 2 8 4 8 16 z 9 > 6 a z + 24 z + ----- + 20 a z + 14 a z + ----- + 31 a z + 2 a a 3 9 10 2 10 > 15 a z + 6 z + 6 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 266]], Vassiliev[3][Knot[11, Alternating, 266]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 266]][q, t]
Out[16]=	17 1 5 1 9 5 13 9 17 -- + 18 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 13 17 17 3 3 2 5 2 5 3 > ----- + ---- + --- + 13 q t + 16 q t + 8 q t + 13 q t + 4 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a266

K11a265 K11a267