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

K11a20 K11a22

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X22,14,1,13 X20,16,21,15 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, -8, 11, -7}

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 18 6 22 20 10 16 14

Alexander Polynomial:

- 5t-2 + 19t-1 - 27 + 19t - 5t2

Conway Polynomial:

1 - z2 - 5z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

q-1 - 2 + 5q - 7q2 + 10q3 - 12q4 + 11q5 - 10q6 + 8q7 - 5q8 + 3q9 - q10

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

{...}

A2 (sl(3)) Invariant:

q-4 + 3q2 - q4 + q6 + 2q8 - 2q10 - 2q14 - q16 - 2q20 + 3q22 + q24 + 2q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 3a-8z2 - 2a-6 - 3a-6z2 - 2a-6z4 - a-4 - 2a-4z2 - 2a-4z4 + a-2 - a-2z4 + 1 + z2

Kauffman Polynomial:

- a-11z + 4a-11z3 - 4a-11z5 + a-11z7 + a-10 - 7a-10z2 + 16a-10z4 - 13a-10z6 + 3a-10z8 + 3a-9z3 + 5a-9z5 - 10a-9z7 + 3a-9z9 + 3a-8 - 22a-8z2 + 42a-8z4 - 29a-8z6 + 4a-8z8 + a-8z10 + 6a-7z - 16a-7z3 + 24a-7z5 - 21a-7z7 + 6a-7z9 + 2a-6 - 16a-6z2 + 28a-6z4 - 21a-6z6 + 4a-6z8 + a-6z10 + 7a-5z - 15a-5z3 + 12a-5z5 - 7a-5z7 + 3a-5z9 - a-4 + 3a-4z2 - 2a-4z4 - 2a-4z6 + 3a-4z8 + 2a-3z - 2a-3z3 - a-3z5 + 3a-3z7 - a-2 + 2a-2z2 - 3a-2z4 + 3a-2z6 - 2a-1z3 + 2a-1z5 + 1 - 2z2 + z4

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

{-1, -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=2 is the signature of 1121. 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 = 21                       1 j = 19                     2   j = 17                   3 1   j = 15                 5 2     j = 13               5 3       j = 11             6 5         j = 9           6 5           j = 7         4 6             j = 5       3 6               j = 3     2 4                 j = 1   1 4                   j = -1   1                     j = -3 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, 21]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 21]]
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[22, 14, 1, 13], X[20, 16, 21, 15], > X[10, 17, 11, 18], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 21]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 21]]
Out[5]=	DTCode[4, 8, 12, 2, 18, 6, 22, 20, 10, 16, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 21]][t]
Out[6]=	5 19 2 -27 - -- + -- + 19 t - 5 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 21]][z]
Out[7]=	2 4 1 - z - 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 21]}
In[9]:=	{KnotDet[Knot[11, Alternating, 21]], KnotSignature[Knot[11, Alternating, 21]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 21]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -2 + - + 5 q - 7 q + 10 q - 12 q + 11 q - 10 q + 8 q - 5 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 21]}
In[12]:=	A2Invariant[Knot[11, Alternating, 21]][q]
Out[12]=	-4 2 4 6 8 10 14 16 20 22 24 q + 3 q - q + q + 2 q - 2 q - 2 q - q - 2 q + 3 q + q + 28 30 32 > 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 21]][a, z]
Out[13]=	2 2 2 4 4 4 -10 3 2 -4 -2 2 3 z 3 z 2 z 2 z 2 z z 1 - a + -- - -- - a + a + z + ---- - ---- - ---- - ---- - ---- - -- 8 6 8 6 4 6 4 2 a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 21]][a, z]
Out[14]=	2 2 -10 3 2 -4 -2 z 6 z 7 z 2 z 2 7 z 22 z 1 + a + -- + -- - a - a - --- + --- + --- + --- - 2 z - ---- - ----- - 8 6 11 7 5 3 10 8 a a a a a a a a 2 2 2 3 3 3 3 3 3 16 z 3 z 2 z 4 z 3 z 16 z 15 z 2 z 2 z 4 > ----- + ---- + ---- + ---- + ---- - ----- - ----- - ---- - ---- + z + 6 4 2 11 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 16 z 42 z 28 z 2 z 3 z 4 z 5 z 24 z 12 z z > ----- + ----- + ----- - ---- - ---- - ---- + ---- + ----- + ----- - -- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 5 6 6 6 6 6 7 7 7 7 2 z 13 z 29 z 21 z 2 z 3 z z 10 z 21 z 7 z > ---- - ----- - ----- - ----- - ---- + ---- + --- - ----- - ----- - ---- + a 10 8 6 4 2 11 9 7 5 a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 3 z 3 z 4 z 4 z 3 z 3 z 6 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, 21]], Vassiliev[3][Knot[11, Alternating, 21]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, Alternating, 21]][q, t]
Out[16]=	3 1 1 q 3 5 5 2 7 2 7 3 4 q + 2 q + ----- + --- + - + 4 q t + 3 q t + 6 q t + 4 q t + 6 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 6 q t + 5 q t + 6 q t + 5 q t + 5 q t + 3 q t + 5 q t + 15 7 17 7 17 8 19 8 21 9 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a21

K11a20 K11a22