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

K11a310 K11a312

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,4,13,3 X14,6,15,5 X20,8,21,7 X18,10,19,9 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, -5, 10, -4, 11, -8}

DT (Dowker-Thistlethwaite) Code:

6 12 14 20 18 4 2 22 10 8 16

Alexander Polynomial:

- 4t-2 + 20t-1 - 31 + 20t - 4t2

Conway Polynomial:

1 + 4z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{79, 2}

Jones Polynomial:

q-1 - 3 + 6q - 9q2 + 12q3 - 12q4 + 12q5 - 10q6 + 7q7 - 4q8 + 2q9 - q10

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-11z + 7a-11z3 - 5a-11z5 + a-11z7 + a-10 - 5a-10z2 + 12a-10z4 - 9a-10z6 + 2a-10z8 + 2a-9z - 4a-9z3 + 7a-9z5 - 7a-9z7 + 2a-9z9 + a-8 - 7a-8z2 + 11a-8z4 - 8a-8z6 + a-8z10 + 6a-7z - 23a-7z3 + 25a-7z5 - 17a-7z7 + 5a-7z9 - 4a-6z2 + 9a-6z4 - 11a-6z6 + 3a-6z8 + a-6z10 + 2a-5z - 3a-5z3 + a-5z5 - 3a-5z7 + 3a-5z9 + 2a-4z2 + 3a-4z4 - 7a-4z6 + 5a-4z8 + 6a-3z3 - 9a-3z5 + 6a-3z7 - a-2 + 3a-2z2 - 6a-2z4 + 5a-2z6 - 3a-1z3 + 3a-1z5 - z2 + z4

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

{4, 10}

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 11311. 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                     1   j = 17                   3 1   j = 15                 4 1     j = 13               6 3       j = 11             6 4         j = 9           6 6           j = 7         6 6             j = 5       3 6               j = 3     3 6                 j = 1   1 4                   j = -1   2                     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, 311]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 311]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[18, 10, 19, 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, 311]]
Out[4]=	GaussCode[1, -7, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -5, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 311]]
Out[5]=	DTCode[6, 12, 14, 20, 18, 4, 2, 22, 10, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 311]][t]
Out[6]=	4 20 2 -31 - -- + -- + 20 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 311]][z]
Out[7]=	2 4 1 + 4 z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 311]}
In[9]:=	{KnotDet[Knot[11, Alternating, 311]], KnotSignature[Knot[11, Alternating, 311]]}
Out[9]=	{79, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 311]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -3 + - + 6 q - 9 q + 12 q - 12 q + 12 q - 10 q + 7 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 311]}
In[12]:=	A2Invariant[Knot[11, Alternating, 311]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 16 20 -1 + q - q + 3 q - 2 q + q + 2 q - q + 2 q - q + q - 2 q + 22 26 28 30 32 > 3 q - q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 311]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 -8 -2 2 2 z z z z z 2 z z -a + a + a + z + ---- + -- - -- + -- - -- - ---- - -- 8 6 4 2 6 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 311]][a, z]
Out[14]=	2 2 2 2 -10 -8 -2 2 z 2 z 6 z 2 z 2 5 z 7 z 4 z 2 z a + a - a - --- + --- + --- + --- - z - ---- - ---- - ---- + ---- + 11 9 7 5 10 8 6 4 a a a a a a a a 2 3 3 3 3 3 3 4 4 3 z 7 z 4 z 23 z 3 z 6 z 3 z 4 12 z 11 z > ---- + ---- - ---- - ----- - ---- + ---- - ---- + z + ----- + ----- + 2 11 9 7 5 3 a 10 8 a a a a a a a a 4 4 4 5 5 5 5 5 5 6 6 9 z 3 z 6 z 5 z 7 z 25 z z 9 z 3 z 9 z 8 z > ---- + ---- - ---- - ---- + ---- + ----- + -- - ---- + ---- - ---- - ---- - 6 4 2 11 9 7 5 3 a 10 8 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 11 z 7 z 5 z z 7 z 17 z 3 z 6 z 2 z 3 z > ----- - ---- + ---- + --- - ---- - ----- - ---- + ---- + ---- + ---- + 6 4 2 11 9 7 5 3 10 6 a a a a a a a a a a 8 9 9 9 10 10 5 z 2 z 5 z 3 z z z > ---- + ---- + ---- + ---- + --- + --- 4 9 7 5 8 6 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 311]], Vassiliev[3][Knot[11, Alternating, 311]]}
Out[15]=	{4, 10}
In[16]:=	Kh[Knot[11, Alternating, 311]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 4 q + 3 q + ----- + --- + - + 6 q t + 3 q t + 6 q t + 6 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 + 6 q t + 6 q t + 4 q t + 6 q t + 3 q t + 4 q t + 15 7 17 7 17 8 19 8 21 9 > q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a311

K11a310 K11a312