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

K11a232 K11a234

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X8,15,9,16 X6,18,7,17 X12,19,13,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 10 20 22 18 2 8 6 12 16

Alexander Polynomial:

t-4 - 6t-3 + 19t-2 - 37t-1 + 47 - 37t + 19t2 - 6t3 + t4

Conway Polynomial:

1 + z2 + 3z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{173, 0}

Jones Polynomial:

- q-5 + 4q-4 - 10q-3 + 18q-2 - 24q-1 + 28 - 28q + 25q2 - 18q3 + 11q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 4q-10 + 3q-8 + 2q-6 - 4q-4 + 6q-2 - 5 + 4q2 - q6 + 5q8 - 5q10 + 2q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - a-2 - 6a-2z2 - 6a-2z4 - 2a-2z6 + 3 + 10z2 + 11z4 + 5z6 + z8 - a2 - 4a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + 4a-5z3 - 9a-5z5 + 5a-5z7 - 2a-4z2 + 13a-4z4 - 21a-4z6 + 10a-4z8 - 2a-3z + 13a-3z3 - 14a-3z5 - 8a-3z7 + 9a-3z9 + a-2 - 14a-2z2 + 48a-2z4 - 62a-2z6 + 21a-2z8 + 3a-2z10 - 5a-1z + 17a-1z3 - 8a-1z5 - 26a-1z7 + 19a-1z9 + 3 - 21z2 + 56z4 - 65z6 + 24z8 + 3z10 - 5az + 17az3 - 16az5 - 4az7 + 10az9 + a2 - 8a2z2 + 18a2z4 - 21a2z6 + 13a2z8 - 2a3z + 8a3z3 - 12a3z5 + 9a3z7 + a4z2 - 4a4z4 + 4a4z6 - a5z3 + a5z5

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

{1, 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=0 is the signature of 11233. 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 = 13                       1 j = 11                     4   j = 9                   7 1   j = 7                 11 4     j = 5               14 7       j = 3             14 11         j = 1           14 14           j = -1         11 15             j = -3       7 13               j = -5     3 11                 j = -7   1 7                   j = -9   3                     j = -11 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, 233]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 233]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[10, 5, 11, 6], X[20, 8, 21, 7], > X[22, 9, 1, 10], X[18, 11, 19, 12], X[2, 14, 3, 13], X[8, 15, 9, 16], > X[6, 18, 7, 17], X[12, 19, 13, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 233]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -8, 5, -3, 6, -10, 7, -2, 8, -11, 9, -6, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 233]]
Out[5]=	DTCode[4, 14, 10, 20, 22, 18, 2, 8, 6, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 233]][t]
Out[6]=	-4 6 19 37 2 3 4 47 + t - -- + -- - -- - 37 t + 19 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 233]][z]
Out[7]=	2 4 6 8 1 + z + 3 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 233]}
In[9]:=	{KnotDet[Knot[11, Alternating, 233]], KnotSignature[Knot[11, Alternating, 233]]}
Out[9]=	{173, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 233]][q]
Out[10]=	-5 4 10 18 24 2 3 4 5 6 28 - q + -- - -- + -- - -- - 28 q + 25 q - 18 q + 11 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 233]}
In[12]:=	A2Invariant[Knot[11, Alternating, 233]][q]
Out[12]=	-14 2 4 3 2 4 6 2 6 8 10 12 -5 - q + --- - --- + -- + -- - -- + -- + 4 q - q + 5 q - 5 q + 2 q - 12 10 8 6 4 2 q q q q q q 16 18 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 233]][a, z]
Out[13]=	2 2 4 4 -2 2 2 z 6 z 2 2 4 z 6 z 2 4 3 - a - a + 10 z + -- - ---- - 4 a z + 11 z + -- - ---- - 3 a z + 4 2 4 2 a a a a 6 6 2 z 2 6 8 > 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 233]][a, z]
Out[14]=	2 2 -2 2 2 z 5 z 3 2 2 z 14 z 2 2 3 + a + a - --- - --- - 5 a z - 2 a z - 21 z - ---- - ----- - 8 a z + 3 a 4 2 a a a 3 3 3 4 4 2 4 z 13 z 17 z 3 3 3 5 3 4 z > a z + ---- + ----- + ----- + 17 a z + 8 a z - a z + 56 z - -- + 5 3 a 6 a a a 4 4 5 5 5 13 z 48 z 2 4 4 4 9 z 14 z 8 z 5 > ----- + ----- + 18 a z - 4 a z - ---- - ----- - ---- - 16 a z - 4 2 5 3 a a a a a 6 6 6 7 3 5 5 5 6 z 21 z 62 z 2 6 4 6 5 z > 12 a z + a z - 65 z + -- - ----- - ----- - 21 a z + 4 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 9 8 z 26 z 7 3 7 8 10 z 21 z 2 8 9 z > ---- - ----- - 4 a z + 9 a z + 24 z + ----- + ----- + 13 a z + ---- + 3 a 4 2 3 a a a a 9 10 19 z 9 10 3 z > ----- + 10 a z + 3 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 233]], Vassiliev[3][Knot[11, Alternating, 233]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 233]][q, t]
Out[16]=	15 1 3 1 7 3 11 7 13 -- + 14 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 11 3 3 2 5 2 5 3 7 3 > --- + 14 q t + 14 q t + 11 q t + 14 q t + 7 q t + 11 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a233

K11a232 K11a234