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

K11a11 K11a13

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X20,12,21,11 X22,14,1,13 X18,16,19,15 X6,17,7,18 X14,20,15,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 20 22 18 6 14 12

Alexander Polynomial:

2t-3 - 11t-2 + 24t-1 - 29 + 24t - 11t2 + 2t3

Conway Polynomial:

1 - 2z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a141, ...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 9 + 13q - 16q2 + 17q3 - 15q4 + 12q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-8 + 2q-4 - 2q-2 - 2 + 2q2 - 3q4 + 3q6 + 2q12 - 3q14 + 3q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + a-4z2 + 2a-4z4 + a-4z6 + 2a-2 + 2a-2z2 + 2a-2z4 + a-2z6 - 3 - 5z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 + 4a-7z3 - 12a-7z5 + 7a-7z7 - a-6z2 + 5a-6z4 - 11a-6z6 + 7a-6z8 + a-5z + a-5z3 - 4a-5z5 - a-5z7 + 4a-5z9 - 4a-4z2 + 15a-4z4 - 18a-4z6 + 8a-4z8 + a-4z10 + 3a-3z - 10a-3z3 + 13a-3z5 - 11a-3z7 + 6a-3z9 - 2a-2 + 4a-2z2 - a-2z4 - 5a-2z6 + 3a-2z8 + a-2z10 + a-1z - a-1z3 - 2a-1z5 - a-1z7 + 2a-1z9 - 3 + 11z2 - 9z4 - z6 + 2z8 - az + 5az3 - 6az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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

{-2, 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 1112. 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 = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 7 3     j = 9               8 5       j = 7             9 7         j = 5           7 8           j = 3         6 9             j = 1       4 8               j = -1     1 5                 j = -3   1 4                   j = -5   1                     j = -7 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, 12]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 12]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[20, 12, 21, 11], X[22, 14, 1, 13], X[18, 16, 19, 15], > X[6, 17, 7, 18], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 12]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -8, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 12]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 20, 22, 18, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 12]][t]
Out[6]=	2 11 24 2 3 -29 + -- - -- + -- + 24 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 12]][z]
Out[7]=	2 4 6 1 - 2 z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 12], Knot[11, Alternating, 141]}
In[9]:=	{KnotDet[Knot[11, Alternating, 12]], KnotSignature[Knot[11, Alternating, 12]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 12]][q]
Out[10]=	-3 2 5 2 3 4 5 6 7 8 -9 + q - -- + - + 13 q - 16 q + 17 q - 15 q + 12 q - 8 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 12]}
In[12]:=	A2Invariant[Knot[11, Alternating, 12]][q]
Out[12]=	-10 -8 2 2 2 4 6 12 14 16 18 -2 + q + q + -- - -- + 2 q - 3 q + 3 q + 2 q - 3 q + 3 q - q - 4 2 q q 20 22 24 > q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 12]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 2 2 z z 2 z 2 2 4 z 2 z 2 z z -3 + -- + 2 a - 5 z - -- + -- + ---- + a z - 2 z - -- + ---- + ---- + -- + 2 6 4 2 6 4 2 4 a a a a a a a a 6 z > -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 12]][a, z]
Out[14]=	2 2 2 2 2 2 z 3 z z 2 z z 4 z 4 z 2 2 -3 - -- - 2 a + -- + --- + - - a z + 11 z + -- - -- - ---- + ---- + 5 a z - 2 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 3 4 4 4 4 z 4 z z 10 z z 3 4 6 z 5 z 15 z z > -- + ---- + -- - ----- - -- + 5 a z - 9 z - ---- + ---- + ----- - -- - 9 7 5 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 5 6 6 2 4 z 12 z 4 z 13 z 2 z 5 6 4 z 11 z > 4 a z + -- - ----- - ---- + ----- - ---- - 6 a z - z + ---- - ----- - 9 7 5 3 a 8 6 a a a a a a 6 6 7 7 7 7 8 18 z 5 z 2 6 7 z z 11 z z 7 8 7 z > ----- - ---- + a z + ---- - -- - ----- - -- + 2 a z + 2 z + ---- + 4 2 7 5 3 a 6 a a a a a a 8 8 9 9 9 10 10 8 z 3 z 4 z 6 z 2 z z z > ---- + ---- + ---- + ---- + ---- + --- + --- 4 2 5 3 a 4 2 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 12]], Vassiliev[3][Knot[11, Alternating, 12]]}
Out[15]=	{-2, 0}
In[16]:=	Kh[Knot[11, Alternating, 12]][q, t]
Out[16]=	3 1 1 1 4 1 5 4 q 3 8 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 7 q t + 8 q t + 9 q t + 7 q t + 8 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a12

K11a11 K11a13