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

K11a308 K11a310

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 X4,12,5,11 X2,14,3,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22

Gauss Code:

{1, -7, 2, -6, 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 4 2 22 10 8 16

Alexander Polynomial:

- 2t-3 + 11t-2 - 21t-1 + 25 - 21t + 11t2 - 2t3

Conway Polynomial:

1 + 5z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a63, ...}

Determinant and Signature:

{93, 4}

Jones Polynomial:

1 - 3q + 6q2 - 9q3 + 13q4 - 14q5 + 15q6 - 13q7 + 9q8 - 6q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q6 - 2q8 + 4q10 + q14 + 2q16 - 2q18 + 2q20 - 2q22 + q26 - 2q28 + q30 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + a-12z2 - 6a-12z4 + 3a-12z6 - 4a-11z + 10a-11z3 - 12a-11z5 + 5a-11z7 + a-10 - 5a-10z2 + 11a-10z4 - 11a-10z6 + 5a-10z8 - 3a-9z + 15a-9z3 - 9a-9z5 - a-9z7 + 3a-9z9 + a-8 - 11a-8z2 + 28a-8z4 - 21a-8z6 + 6a-8z8 + a-8z10 + a-7z - 5a-7z3 + 14a-7z5 - 14a-7z7 + 6a-7z9 + a-6 - 13a-6z2 + 23a-6z4 - 19a-6z6 + 5a-6z8 + a-6z10 - 2a-5z3 + a-5z5 - 5a-5z7 + 3a-5z9 + 2a-4 - 6a-4z2 + 9a-4z4 - 11a-4z6 + 4a-4z8 + 6a-3z3 - 9a-3z5 + 3a-3z7 + 2a-2z2 - 3a-2z4 + a-2z6

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

{5, 12}

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 11309. 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                 5 2     j = 15               8 4       j = 13             7 5         j = 11           7 8           j = 9         6 7             j = 7       3 7               j = 5     3 6                 j = 3   1 4                   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, 309]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 309]]
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[4, 12, 5, 11], X[2, 14, 3, 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, 309]]
Out[4]=	GaussCode[1, -7, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -4, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 309]]
Out[5]=	DTCode[6, 12, 14, 18, 20, 4, 2, 22, 10, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 309]][t]
Out[6]=	2 11 21 2 3 25 - -- + -- - -- - 21 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 309]][z]
Out[7]=	2 4 6 1 + 5 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 63], Knot[11, Alternating, 309]}
In[9]:=	{KnotDet[Knot[11, Alternating, 309]], KnotSignature[Knot[11, Alternating, 309]]}
Out[9]=	{93, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 309]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 6 q - 9 q + 13 q - 14 q + 15 q - 13 q + 9 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, 309]}
In[12]:=	A2Invariant[Knot[11, Alternating, 309]][q]
Out[12]=	2 6 8 10 14 16 18 20 22 26 1 - q + q - 2 q + 4 q + q + 2 q - 2 q + 2 q - 2 q + q - 28 30 34 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 309]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 -10 -8 -6 2 z 4 z z z 2 z 2 z 2 z 2 z -a + a - a + -- - --- + ---- - -- + -- + ---- + ---- - ---- - ---- + 4 10 8 6 4 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, 309]][a, z]
Out[14]=	2 2 2 2 2 -10 -8 -6 2 4 z 3 z z z 5 z 11 z 13 z 6 z a + a + a + -- - --- - --- + -- + --- - ---- - ----- - ----- - ---- + 4 11 9 7 12 10 8 6 4 a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 2 z 2 z 10 z 15 z 5 z 2 z 6 z 6 z 11 z 28 z > ---- - ---- + ----- + ----- - ---- - ---- + ---- - ---- + ----- + ----- + 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 23 z 9 z 3 z z 12 z 9 z 14 z z 9 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 11 z 21 z 19 z 11 z z 5 z z 14 z 5 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 5 z 6 z 5 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, 309]], Vassiliev[3][Knot[11, Alternating, 309]]}
Out[15]=	{5, 12}
In[16]:=	Kh[Knot[11, Alternating, 309]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 4 q + 3 q + ---- + --- + -- + 6 q t + 3 q t + 7 q t + 6 q t + 7 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 7 q t + 8 q t + 7 q t + 5 q t + 8 q t + 4 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 5 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 K11a309

K11a308 K11a310