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

K11a311 K11a313

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 14 18 20 2 22 4 10 8

Alexander Polynomial:

3t-3 - 13t-2 + 27t-1 - 33 + 27t - 13t2 + 3t3

Conway Polynomial:

1 + 2z2 + 5z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{119, -2}

Jones Polynomial:

q-9 - 4q-8 + 8q-7 - 13q-6 + 17q-5 - 19q-4 + 19q-3 - 16q-2 + 12q-1 - 6 + 3q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - 2q-24 + 2q-22 - 3q-20 - q-18 + 2q-16 - 3q-14 + 4q-12 - q-10 + 2q-8 + 2q-6 - 3q-4 + 4q-2 - 1 + q4 - q6

HOMFLY-PT Polynomial:

- 2z2 - z4 + a2z2 + 2a2z4 + a2z6 + 4a4 + 9a4z2 + 7a4z4 + 2a4z6 - 4a6 - 7a6z2 - 3a6z4 + a8 + a8z2

Kauffman Polynomial:

- 2a-1z3 + a-1z5 + 2z2 - 6z4 + 3z6 + 4az3 - 9az5 + 5az7 - 7a2z2 + 20a2z4 - 17a2z6 + 7a2z8 + 2a3z - 6a3z3 + 16a3z5 - 13a3z7 + 6a3z9 + 4a4 - 25a4z2 + 48a4z4 - 33a4z6 + 8a4z8 + 2a4z10 + 2a5z - 16a5z3 + 31a5z5 - 28a5z7 + 11a5z9 + 4a6 - 18a6z2 + 29a6z4 - 27a6z6 + 7a6z8 + 2a6z10 + 2a7z3 - 5a7z5 - 6a7z7 + 5a7z9 + a8 - a8z2 + 5a8z4 - 13a8z6 + 6a8z8 + 6a9z3 - 10a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

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

{2, -2}

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 11312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     2   j = 1                   4 1   j = -1                 8 2     j = -3               9 5       j = -5             10 7         j = -7           9 9           j = -9         8 10             j = -11       5 9               j = -13     3 8                 j = -15   1 5                   j = -17   3                     j = -19 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, 312]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 312]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[4, 17, 5, 18], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 312]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -1, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 312]]
Out[5]=	DTCode[6, 12, 16, 14, 18, 20, 2, 22, 4, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 312]][t]
Out[6]=	3 13 27 2 3 -33 + -- - -- + -- + 27 t - 13 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 312]][z]
Out[7]=	2 4 6 1 + 2 z + 5 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 312]}
In[9]:=	{KnotDet[Knot[11, Alternating, 312]], KnotSignature[Knot[11, Alternating, 312]]}
Out[9]=	{119, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 312]][q]
Out[10]=	-9 4 8 13 17 19 19 16 12 2 -6 + q - -- + -- - -- + -- - -- + -- - -- + -- + 3 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 312]}
In[12]:=	A2Invariant[Knot[11, Alternating, 312]][q]
Out[12]=	-28 2 2 3 -18 2 3 4 -10 2 2 3 -1 + q - --- + --- - --- - q + --- - --- + --- - q + -- + -- - -- + 24 22 20 16 14 12 8 6 4 q q q q q q q q q 4 4 6 > -- + q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 312]][a, z]
Out[13]=	4 6 8 2 2 2 4 2 6 2 8 2 4 2 4 4 a - 4 a + a - 2 z + a z + 9 a z - 7 a z + a z - z + 2 a z + 4 4 6 4 2 6 4 6 > 7 a z - 3 a z + a z + 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 312]][a, z]
Out[14]=	4 6 8 3 5 2 2 2 4 2 6 2 4 a + 4 a + a + 2 a z + 2 a z + 2 z - 7 a z - 25 a z - 18 a z - 3 8 2 10 2 2 z 3 3 3 5 3 7 3 9 3 > a z + a z - ---- + 4 a z - 6 a z - 16 a z + 2 a z + 6 a z - a 5 4 2 4 4 4 6 4 8 4 10 4 z 5 > 6 z + 20 a z + 48 a z + 29 a z + 5 a z - 2 a z + -- - 9 a z + a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 16 a z + 31 a z - 5 a z - 10 a z + 3 z - 17 a z - 33 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 27 a z - 13 a z + a z + 5 a z - 13 a z - 28 a z - 6 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 4 a z + 7 a z + 8 a z + 7 a z + 6 a z + 6 a z + 11 a z + 7 9 4 10 6 10 > 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 312]], Vassiliev[3][Knot[11, Alternating, 312]]}
Out[15]=	{2, -2}
In[16]:=	Kh[Knot[11, Alternating, 312]][q, t]
Out[16]=	5 8 1 3 1 5 3 8 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 9 8 10 9 9 10 7 9 2 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a312

K11a311 K11a313