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

K11a307 K11a309

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 18 20 2 4 22 10 8 16

Alexander Polynomial:

- t-4 + 5t-3 - 9t-2 + 13t-1 - 15 + 13t - 9t2 + 5t3 - t4

Conway Polynomial:

1 + 6z2 + z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{71, 6}

Jones Polynomial:

q - 2q2 + 4q3 - 6q4 + 9q5 - 10q6 + 11q7 - 10q8 + 8q9 - 6q10 + 3q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q4 + q8 - q12 + 2q14 - q16 + 3q18 + q24 - 2q26 + q28 - q30 - q36

HOMFLY-PT Polynomial:

- 2a-10 - 3a-10z2 - a-10z4 + 3a-8 + 11a-8z2 + 9a-8z4 + 2a-8z6 - 2a-6 - 9a-6z2 - 12a-6z4 - 6a-6z6 - a-6z8 + 2a-4 + 7a-4z2 + 5a-4z4 + a-4z6

Kauffman Polynomial:

a-15z3 + 3a-14z4 + 2a-13z - 5a-13z3 + 6a-13z5 + 4a-12z2 - 11a-12z4 + 8a-12z6 - 2a-11z + 5a-11z3 - 14a-11z5 + 8a-11z7 + 2a-10 - 4a-10z2 + 2a-10z4 - 11a-10z6 + 6a-10z8 - 4a-9z + 17a-9z3 - 18a-9z5 - a-9z7 + 3a-9z9 + 3a-8 - 13a-8z2 + 30a-8z4 - 25a-8z6 + 4a-8z8 + a-8z10 - 4a-7z3 + 21a-7z5 - 20a-7z7 + 5a-7z9 + 2a-6 - 14a-6z2 + 26a-6z4 - 12a-6z6 - a-6z8 + a-6z10 - 10a-5z3 + 19a-5z5 - 11a-5z7 + 2a-5z9 + 2a-4 - 9a-4z2 + 12a-4z4 - 6a-4z6 + a-4z8

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

{6, 16}

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=6 is the signature of 11308. 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 = 25                       1 j = 23                     2   j = 21                   4 1   j = 19                 4 2     j = 17               6 4       j = 15             5 4         j = 13           5 6           j = 11         4 5             j = 9       2 5               j = 7     2 4                 j = 5   1 3                   j = 3   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, 308]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 308]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[14, 6, 15, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[2, 12, 3, 11], X[4, 14, 5, 13], X[22, 15, 1, 16], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 308]]
Out[4]=	GaussCode[1, -6, 2, -7, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -4, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 308]]
Out[5]=	DTCode[6, 12, 14, 18, 20, 2, 4, 22, 10, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 308]][t]
Out[6]=	-4 5 9 13 2 3 4 -15 - t + -- - -- + -- + 13 t - 9 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 308]][z]
Out[7]=	2 4 6 8 1 + 6 z + z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 308]}
In[9]:=	{KnotDet[Knot[11, Alternating, 308]], KnotSignature[Knot[11, Alternating, 308]]}
Out[9]=	{71, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 308]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 6 q + 9 q - 10 q + 11 q - 10 q + 8 q - 6 q + 3 q - 12 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 308]}
In[12]:=	A2Invariant[Knot[11, Alternating, 308]][q]
Out[12]=	4 8 12 14 16 18 24 26 28 30 36 q + q - q + 2 q - q + 3 q + q - 2 q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 308]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -2 3 2 2 3 z 11 z 9 z 7 z z 9 z 12 z 5 z --- + -- - -- + -- - ---- + ----- - ---- + ---- - --- + ---- - ----- + ---- + 10 8 6 4 10 8 6 4 10 8 6 4 a a a a a a a a a a a a 6 6 6 8 2 z 6 z z z > ---- - ---- + -- - -- 8 6 4 6 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 308]][a, z]
Out[14]=	2 2 2 2 2 2 3 2 2 2 z 2 z 4 z 4 z 4 z 13 z 14 z 9 z --- + -- + -- + -- + --- - --- - --- + ---- - ---- - ----- - ----- - ---- + 10 8 6 4 13 11 9 12 10 8 6 4 a a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 z 5 z 5 z 17 z 4 z 10 z 3 z 11 z 2 z 30 z > --- - ---- + ---- + ----- - ---- - ----- + ---- - ----- + ---- + ----- + 15 13 11 9 7 5 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 26 z 12 z 6 z 14 z 18 z 21 z 19 z 8 z 11 z > ----- + ----- + ---- - ----- - ----- + ----- + ----- + ---- - ----- - 6 4 13 11 9 7 5 12 10 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 8 25 z 12 z 6 z 8 z z 20 z 11 z 6 z 4 z z z > ----- - ----- - ---- + ---- - -- - ----- - ----- + ---- + ---- - -- + -- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 9 9 9 10 10 3 z 5 z 2 z z z > ---- + ---- + ---- + --- + --- 9 7 5 8 6 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 308]], Vassiliev[3][Knot[11, Alternating, 308]]}
Out[15]=	{6, 16}
In[16]:=	Kh[Knot[11, Alternating, 308]][q, t]
Out[16]=	3 5 5 7 q q q 7 9 9 2 11 2 11 3 3 q + 2 q + -- + -- + -- + 4 q t + 2 q t + 5 q t + 4 q t + 5 q t + 2 t t t 13 3 13 4 15 4 15 5 17 5 17 6 > 5 q t + 6 q t + 5 q t + 4 q t + 6 q t + 4 q t + 19 6 19 7 21 7 21 8 23 8 25 9 > 4 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 K11a308

K11a307 K11a309