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

K11a211 K11a213

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 22 18 2 8 6 20 10 14

Alexander Polynomial:

- 3t-3 + 16t-2 - 35t-1 + 45 - 35t + 16t2 - 3t3

Conway Polynomial:

1 + 2z2 - 2z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{153, 4}

Jones Polynomial:

1 - 3q + 8q2 - 14q3 + 21q4 - 24q5 + 25q6 - 23q7 + 17q8 - 11q9 + 5q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 4q8 + 5q10 - q12 - q14 + 4q16 - 4q18 + 4q20 - 4q22 - q24 + 2q26 - 4q28 + 3q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + 4a-8z2 + 3a-8z4 - a-6 - 4a-6z2 - 5a-6z4 - 2a-6z6 + a-4 + a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

a-13z5 - 4a-12z4 + 5a-12z6 + 5a-11z3 - 15a-11z5 + 11a-11z7 - a-10z2 + 4a-10z4 - 17a-10z6 + 13a-10z8 - a-9z + 17a-9z3 - 28a-9z5 + 4a-9z7 + 8a-9z9 - 7a-8z2 + 31a-8z4 - 48a-8z6 + 21a-8z8 + 2a-8z10 - 5a-7z + 19a-7z3 - 14a-7z5 - 13a-7z7 + 13a-7z9 + a-6 - 11a-6z2 + 31a-6z4 - 36a-6z6 + 13a-6z8 + 2a-6z10 - 5a-5z + 12a-5z3 - 9a-5z5 - 3a-5z7 + 5a-5z9 + a-4 - 2a-4z2 + 5a-4z4 - 9a-4z6 + 5a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{2, 3}

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=4 is the signature of 11212. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     4   j = 19                   7 1   j = 17                 10 4     j = 15               13 7       j = 13             12 10         j = 11           12 13           j = 9         9 12             j = 7       5 12               j = 5     3 9                 j = 3   1 6                   j = 1   2                     j = -1 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, 212]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 212]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[8, 14, 9, 13], X[6, 15, 7, 16], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 212]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -7, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, > -9, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 212]]
Out[5]=	DTCode[4, 12, 16, 22, 18, 2, 8, 6, 20, 10, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 212]][t]
Out[6]=	3 16 35 2 3 45 - -- + -- - -- - 35 t + 16 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 212]][z]
Out[7]=	2 4 6 1 + 2 z - 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 212]}
In[9]:=	{KnotDet[Knot[11, Alternating, 212]], KnotSignature[Knot[11, Alternating, 212]]}
Out[9]=	{153, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 212]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 8 q - 14 q + 21 q - 24 q + 25 q - 23 q + 17 q - 11 q + 10 11 > 5 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 212]}
In[12]:=	A2Invariant[Knot[11, Alternating, 212]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 4 q + 5 q - q - q + 4 q - 4 q + 4 q - 4 q - 24 26 28 30 32 34 > q + 2 q - 4 q + 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 212]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -6 -4 -2 z 4 z 4 z z 2 z 3 z 5 z z z -a + a + a - --- + ---- - ---- + -- + ---- + ---- - ---- - -- + -- - 10 8 6 4 2 8 6 4 2 a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 212]][a, z]
Out[14]=	2 2 2 2 2 -6 -4 -2 z 5 z 5 z z z 7 z 11 z 2 z 3 z a + a - a - -- - --- - --- - -- - --- - ---- - ----- - ---- + ---- + 9 7 5 3 10 8 6 4 2 a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 5 z 17 z 19 z 12 z 5 z 4 z 4 z 31 z 31 z 5 z > ---- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + ----- + ---- - 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 4 5 5 5 5 5 5 6 6 6 3 z z 15 z 28 z 14 z 9 z 7 z 5 z 17 z 48 z > ---- + --- - ----- - ----- - ----- - ---- - ---- + ---- - ----- - ----- - 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 36 z 9 z z 11 z 4 z 13 z 3 z 3 z 13 z 21 z > ----- - ---- + -- + ----- + ---- - ----- - ---- + ---- + ----- + ----- + 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a 8 8 9 9 9 10 10 13 z 5 z 8 z 13 z 5 z 2 z 2 z > ----- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 9 7 5 8 6 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 212]], Vassiliev[3][Knot[11, Alternating, 212]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, Alternating, 212]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 9 q t + 5 q t + 12 q t + 9 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 12 q t + 12 q t + 13 q t + 12 q t + 10 q t + 13 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 10 q t + 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a212

K11a211 K11a213