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

K11a203 K11a205

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 18 20 2 22 10 8 6 14

Alexander Polynomial:

- 2t-3 + 11t-2 - 23t-1 + 29 - 23t + 11t2 - 2t3

Conway Polynomial:

1 + 3z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a222, ...}

Determinant and Signature:

{101, 4}

Jones Polynomial:

1 - 3q + 7q2 - 10q3 + 14q4 - 16q5 + 16q6 - 14q7 + 10q8 - 6q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 3a-12z6 - 3a-11z + 9a-11z3 - 11a-11z5 + 5a-11z7 + a-10 - 3a-10z2 + 8a-10z4 - 9a-10z6 + 5a-10z8 - 4a-9z + 15a-9z3 - 11a-9z5 + a-9z7 + 3a-9z9 + 2a-8 - 10a-8z2 + 22a-8z4 - 19a-8z6 + 7a-8z8 + a-8z10 - 2a-7z + 2a-7z3 + 3a-7z5 - 9a-7z7 + 6a-7z9 + 2a-6 - 7a-6z2 + 12a-6z4 - 16a-6z6 + 6a-6z8 + a-6z10 - a-5z + 3a-5z3 - 6a-5z5 - 2a-5z7 + 3a-5z9 + a-4 + a-4z2 + a-4z4 - 8a-4z6 + 4a-4z8 + 5a-3z3 - 8a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{3, 7}

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 11204. 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                 6 2     j = 15               8 4       j = 13             8 6         j = 11           8 8           j = 9         6 8             j = 7       4 8               j = 5     3 6                 j = 3   1 5                   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, 204]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 204]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[10, 16, 11, 15], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 204]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -8, 6, -2, 7, -11, 8, -3, 9, -4, 10, > -5, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 204]]
Out[5]=	DTCode[4, 12, 16, 18, 20, 2, 22, 10, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 204]][t]
Out[6]=	2 11 23 2 3 29 - -- + -- - -- - 23 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 204]][z]
Out[7]=	2 4 6 1 + 3 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 204], Knot[11, Alternating, 222]}
In[9]:=	{KnotDet[Knot[11, Alternating, 204]], KnotSignature[Knot[11, Alternating, 204]]}
Out[9]=	{101, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 204]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 10 q + 14 q - 16 q + 16 q - 14 q + 10 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, 204]}
In[12]:=	A2Invariant[Knot[11, Alternating, 204]][q]
Out[12]=	2 4 6 8 10 12 16 18 20 22 24 1 - q + q + 2 q - 2 q + 4 q - q + q - 3 q + 2 q - 2 q + q + 26 28 30 34 > 2 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 204]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 2 2 -4 -2 z 4 z 2 z 2 z 2 z 2 z 2 z -a + -- - -- + a + a - --- + ---- - ---- + ---- + ---- - ---- - ---- + 8 6 10 8 6 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, 204]][a, z]
Out[14]=	2 2 2 -10 2 2 -4 -2 3 z 4 z 2 z z 2 z 3 z 10 z a + -- + -- + a - a - --- - --- - --- - -- + ---- - ---- - ----- - 8 6 11 9 7 5 12 10 8 a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 4 7 z z 3 z 2 z 9 z 15 z 2 z 3 z 5 z 6 z 8 z > ---- + -- + ---- - ---- + ---- + ----- + ---- + ---- + ---- - ---- + ---- + 6 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 22 z 12 z z 3 z z 11 z 11 z 3 z 6 z 8 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 7 3 z 9 z 19 z 16 z 8 z z 5 z z 9 z 2 z 3 z > ---- - ---- - ----- - ----- - ---- + -- + ---- + -- - ---- - ---- + ---- + 12 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a a 8 8 8 8 9 9 9 10 10 5 z 7 z 6 z 4 z 3 z 6 z 3 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, 204]], Vassiliev[3][Knot[11, Alternating, 204]]}
Out[15]=	{3, 7}
In[16]:=	Kh[Knot[11, Alternating, 204]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 5 q + 3 q + ---- + --- + -- + 6 q t + 4 q t + 8 q t + 6 q t + 8 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 8 q t + 8 q t + 8 q t + 6 q t + 8 q t + 4 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 6 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 K11a204

K11a203 K11a205