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

K11a19 K11a21

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X20,14,21,13 X22,16,1,15 X10,17,11,18 X16,20,17,19 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 18 6 20 22 10 16 14

Alexander Polynomial:

- 3t-3 + 13t-2 - 25t-1 + 31 - 25t + 13t2 - 3t3

Conway Polynomial:

1 - 5z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{113, 4}

Jones Polynomial:

1 - 2q + 6q2 - 10q3 + 15q4 - 18q5 + 18q6 - 17q7 + 13q8 - 8q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 4a-8 + 7a-8z2 + 3a-8z4 - 5a-6 - 9a-6z2 - 7a-6z4 - 2a-6z6 + a-4 - 2a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 4a-12z6 - 2a-11z + 5a-11z3 - 11a-11z5 + 7a-11z7 + a-10 + 2a-10z2 - 3a-10z4 - 7a-10z6 + 7a-10z8 - 7a-9z + 22a-9z3 - 23a-9z5 + 5a-9z7 + 4a-9z9 + 4a-8 - 15a-8z2 + 30a-8z4 - 30a-8z6 + 12a-8z8 + a-8z10 - 7a-7z + 14a-7z3 - 5a-7z5 - 8a-7z7 + 7a-7z9 + 5a-6 - 22a-6z2 + 34a-6z4 - 26a-6z6 + 8a-6z8 + a-6z10 - a-5z + a-5z5 - 4a-5z7 + 3a-5z9 + a-4 - 2a-4z2 + 3a-4z4 - 6a-4z6 + 3a-4z8 + a-3z + 2a-3z3 - 5a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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=4 is the signature of 1120. 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                     3   j = 19                   5 1   j = 17                 8 3     j = 15               9 5       j = 13             9 8         j = 11           9 9           j = 9         6 9             j = 7       4 9               j = 5     2 6                 j = 3   1 5                   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, 20]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 20]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[6, 12, 7, 11], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 20]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 20]]
Out[5]=	DTCode[4, 8, 12, 2, 18, 6, 20, 22, 10, 16, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 20]][t]
Out[6]=	3 13 25 2 3 31 - -- + -- - -- - 25 t + 13 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 20]][z]
Out[7]=	4 6 1 - 5 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 20]}
In[9]:=	{KnotDet[Knot[11, Alternating, 20]], KnotSignature[Knot[11, Alternating, 20]]}
Out[9]=	{113, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 20]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 6 q - 10 q + 15 q - 18 q + 18 q - 17 q + 13 q - 8 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 20]}
In[12]:=	A2Invariant[Knot[11, Alternating, 20]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 1 + q + 3 q - 2 q + 4 q - q - 2 q + q - 5 q + 2 q - 2 q + 24 26 28 30 34 > q + 3 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 20]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 4 5 -4 2 z 7 z 9 z 3 z 3 z 7 z 2 z -a + -- - -- + a + -- - --- + ---- - ---- + ---- + ---- - ---- - ---- + 8 6 2 10 8 6 2 8 6 4 a a a a a a a a a a 4 6 6 z 2 z z > -- - ---- - -- 2 6 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 20]][a, z]
Out[14]=	2 2 2 -10 4 5 -4 2 2 z 7 z 7 z z z 2 z 2 z 15 z a + -- + -- + a - -- - --- - --- - --- - -- + -- + ---- + ---- - ----- - 8 6 2 11 9 7 5 3 12 10 8 a a a a a a a a a a a 2 2 2 3 3 3 3 3 4 4 22 z 2 z 5 z z 5 z 22 z 14 z 2 z 6 z 3 z > ----- - ---- + ---- - --- + ---- + ----- + ----- + ---- - ---- - ---- + 6 4 2 13 11 9 7 3 12 10 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 30 z 34 z 3 z 4 z z 11 z 23 z 5 z z 5 z > ----- + ----- + ---- - ---- + --- - ----- - ----- - ---- + -- - ---- + 8 6 4 2 13 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 4 z 7 z 30 z 26 z 6 z z 7 z 5 z 8 z 4 z > ---- - ---- - ----- - ----- - ---- + -- + ---- + ---- - ---- - ---- + 12 10 8 6 4 2 11 9 7 5 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 2 z 7 z 12 z 8 z 3 z 4 z 7 z 3 z z z > ---- + ---- + ----- + ---- + ---- + ---- + ---- + ---- + --- + --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 20]], Vassiliev[3][Knot[11, Alternating, 20]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 20]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 5 q + 2 q + ---- + - + -- + 6 q t + 4 q t + 9 q t + 6 q t + 9 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 9 q t + 9 q t + 9 q t + 8 q t + 9 q t + 5 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a20

K11a19 K11a21