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

K11a303 K11a305

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X20,6,21,5 X14,8,15,7 X2,10,3,9 X18,11,19,12 X8,14,9,13 X22,16,1,15 X12,17,13,18 X4,20,5,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 20 14 2 18 8 22 12 4 16

Alexander Polynomial:

- 3t-3 + 14t-2 - 26t-1 + 31 - 26t + 14t2 - 3t3

Conway Polynomial:

1 + 3z2 - 4z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{117, 4}

Jones Polynomial:

1 - 3q + 8q2 - 12q3 + 16q4 - 19q5 + 19q6 - 16q7 + 12q8 - 7q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-13z - 2a-13z3 + a-13z5 + a-12z2 - 5a-12z4 + 3a-12z6 - a-11z + a-11z3 - 7a-11z5 + 5a-11z7 + 2a-10 - 11a-10z2 + 17a-10z4 - 15a-10z6 + 7a-10z8 - 2a-9z + 3a-9z3 + 8a-9z5 - 11a-9z7 + 6a-9z9 + 5a-8 - 29a-8z2 + 57a-8z4 - 39a-8z6 + 9a-8z8 + 2a-8z10 - 4a-7z3 + 25a-7z5 - 28a-7z7 + 11a-7z9 + 5a-6 - 26a-6z2 + 44a-6z4 - 33a-6z6 + 7a-6z8 + 2a-6z10 - a-5z3 + 2a-5z5 - 9a-5z7 + 5a-5z9 + 2a-4 - 6a-4z2 + 6a-4z4 - 11a-4z6 + 5a-4z8 + 3a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-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 11304. 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                   5 1   j = 17                 7 2     j = 15               9 5       j = 13             10 7         j = 11           9 9           j = 9         7 10             j = 7       5 9               j = 5     3 7                 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, 304]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 304]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[20, 6, 21, 5], X[14, 8, 15, 7], > X[2, 10, 3, 9], X[18, 11, 19, 12], X[8, 14, 9, 13], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[4, 20, 5, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 304]]
Out[4]=	GaussCode[1, -5, 2, -10, 3, -1, 4, -7, 5, -2, 6, -9, 7, -4, 8, -11, 9, -6, 10, > -3, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 304]]
Out[5]=	DTCode[6, 10, 20, 14, 2, 18, 8, 22, 12, 4, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 304]][t]
Out[6]=	3 14 26 2 3 31 - -- + -- - -- - 26 t + 14 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 304]][z]
Out[7]=	2 4 6 1 + 3 z - 4 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 304]}
In[9]:=	{KnotDet[Knot[11, Alternating, 304]], KnotSignature[Knot[11, Alternating, 304]]}
Out[9]=	{117, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 304]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 8 q - 12 q + 16 q - 19 q + 19 q - 16 q + 12 q - 7 q + 3 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 304]}
In[12]:=	A2Invariant[Knot[11, Alternating, 304]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 2 q + 4 q - 2 q - q + q - 4 q + 3 q - 2 q + 24 26 28 30 32 34 > 2 q + 3 q - 2 q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 304]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 -2 5 5 2 -2 z 8 z 9 z 3 z 2 z 3 z 7 z z --- + -- - -- + -- + a - --- + ---- - ---- + ---- + ---- + ---- - ---- - -- + 10 8 6 4 10 8 6 4 2 8 6 4 a a 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, 304]][a, z]
Out[14]=	2 2 2 2 2 5 5 2 -2 z z 2 z z 11 z 29 z 26 z --- + -- + -- + -- - a + --- - --- - --- + --- - ----- - ----- - ----- - 10 8 6 4 13 11 9 12 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 4 6 z 3 z 2 z z 3 z 4 z z 3 z 5 z 17 z 57 z > ---- + ---- - ---- + --- + ---- - ---- - -- + ---- - ---- + ----- + ----- + 4 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 44 z 6 z 3 z z 7 z 8 z 25 z 2 z 7 z 3 z > ----- + ---- - ---- + --- - ---- + ---- + ----- + ---- - ---- + ---- - 6 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 15 z 39 z 33 z 11 z z 5 z 11 z 28 z 9 z 3 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 7 z 9 z 7 z 5 z 6 z 11 z 5 z 2 z 2 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, 304]], Vassiliev[3][Knot[11, Alternating, 304]]}
Out[15]=	{3, 8}
In[16]:=	Kh[Knot[11, Alternating, 304]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 7 q t + 5 q t + 9 q t + 7 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 10 q t + 9 q t + 9 q t + 10 q t + 7 q t + 9 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 5 q t + 7 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a304

K11a303 K11a305