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

K11a219 K11a221

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 14 2 10 22 8 6 16

Alexander Polynomial:

- 2t-3 + 10t-2 - 19t-1 + 23 - 19t + 10t2 - 2t3

Conway Polynomial:

1 + 3z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{85, 4}

Jones Polynomial:

1 - 2q + 5q2 - 8q3 + 11q4 - 13q5 + 14q6 - 12q7 + 9q8 - 6q9 + 3q10 - q11

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

{K11a60, ...}

A2 (sl(3)) Invariant:

1 + q4 + 2q6 - 2q8 + 2q10 - 2q12 - q14 + 2q16 - q18 + 3q20 - q22 + q26 - 2q28 + q30 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + a-12z2 - 6a-12z4 + 3a-12z6 - 3a-11z + 9a-11z3 - 12a-11z5 + 5a-11z7 + a-10 - 6a-10z2 + 13a-10z4 - 12a-10z6 + 5a-10z8 - 2a-9z + 9a-9z3 - 3a-9z5 - 3a-9z7 + 3a-9z9 + a-8 - 9a-8z2 + 22a-8z4 - 15a-8z6 + 4a-8z8 + a-8z10 + 2a-7z - 11a-7z3 + 18a-7z5 - 13a-7z7 + 5a-7z9 - a-6 + a-6z2 - 3a-6z6 + a-6z8 + a-6z10 + a-5z - 5a-5z3 + 2a-5z5 - 3a-5z7 + 2a-5z9 - 2a-4 + 8a-4z2 - 7a-4z4 - 2a-4z6 + 2a-4z8 + 4a-3z3 - 6a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{3, 8}

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 11220. 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                     2   j = 19                   4 1   j = 17                 5 2     j = 15               7 4       j = 13             7 5         j = 11           6 7           j = 9         5 7             j = 7       3 6               j = 5     2 5                 j = 3   1 4                   j = 1   1                     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, 220]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 220]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[14, 10, 15, 9], X[2, 12, 3, 11], X[10, 14, 11, 13], X[22, 15, 1, 16], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 220]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -7, 6, -2, 7, -5, 8, -11, 9, -3, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 220]]
Out[5]=	DTCode[4, 12, 18, 20, 14, 2, 10, 22, 8, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 220]][t]
Out[6]=	2 10 19 2 3 23 - -- + -- - -- - 19 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 220]][z]
Out[7]=	2 4 6 1 + 3 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 220]}
In[9]:=	{KnotDet[Knot[11, Alternating, 220]], KnotSignature[Knot[11, Alternating, 220]]}
Out[9]=	{85, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 220]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 8 q + 11 q - 13 q + 14 q - 12 q + 9 q - 6 q + 3 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 60], Knot[11, Alternating, 220]}
In[12]:=	A2Invariant[Knot[11, Alternating, 220]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 26 1 + q + 2 q - 2 q + 2 q - 2 q - q + 2 q - q + 3 q - q + q - 28 30 34 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 220]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 -8 -6 2 2 z 4 z 3 z 3 z 2 z 2 z 3 z -a + a + a - -- + -- - --- + ---- - ---- + ---- + ---- - ---- - ---- + 4 2 10 8 4 2 8 6 4 a a a a a a a a a 4 6 6 z z z > -- - -- - -- 2 6 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 220]][a, z]
Out[14]=	2 2 2 2 -10 -8 -6 2 2 3 z 2 z 2 z z z 6 z 9 z z a + a - a - -- - -- - --- - --- + --- + -- + --- - ---- - ---- + -- + 4 2 11 9 7 5 12 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 8 z 5 z 2 z 9 z 9 z 11 z 5 z 4 z 6 z 13 z > ---- + ---- - ---- + ---- + ---- - ----- - ---- + ---- - ---- + ----- + 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 22 z 7 z 4 z z 12 z 3 z 18 z 2 z 6 z 3 z > ----- - ---- - ---- + --- - ----- - ---- + ----- + ---- - ---- + ---- - 8 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 12 z 15 z 3 z 2 z z 5 z 3 z 13 z 3 z 2 z > ----- - ----- - ---- - ---- + -- + ---- - ---- - ----- - ---- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 5 z 4 z z 2 z 3 z 5 z 2 z z z > ---- + ---- + -- + ---- + ---- + ---- + ---- + --- + --- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 220]], Vassiliev[3][Knot[11, Alternating, 220]]}
Out[15]=	{3, 8}
In[16]:=	Kh[Knot[11, Alternating, 220]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 5 q t + 3 q t + 6 q t + 5 q t + 7 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 7 q t + 7 q t + 5 q t + 7 q t + 4 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 5 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a220

K11a219 K11a221