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

K11a312 K11a314

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,3,13,4 X16,5,17,6 X14,8,15,7 X20,9,21,10 X18,11,19,12 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, -6, 10, -5, 11, -8}

DT (Dowker-Thistlethwaite) Code:

6 12 16 14 20 18 2 22 4 10 8

Alexander Polynomial:

5t-2 - 19t-1 + 29 - 19t + 5t2

Conway Polynomial:

1 + z2 + 5z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{77, 0}

Jones Polynomial:

q-8 - 3q-7 + 5q-6 - 8q-5 + 10q-4 - 11q-3 + 12q-2 - 10q-1 + 8 - 5q + 3q2 - q3

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

{...}

A2 (sl(3)) Invariant:

q-26 + q-24 - 2q-22 - q-18 - 3q-16 + 2q-14 + q-10 + 2q-8 + 2q-4 - q-2 + 2q2 - 2q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z4 + a2 + 2a2z2 + 2a2z4 + 2a4 + 3a4z2 + 2a4z4 - 3a6 - 3a6z2 + a8

Kauffman Polynomial:

a-3z3 - a-2z2 + 3a-2z4 - 3a-1z3 + 5a-1z5 + 2z2 - 9z4 + 7z6 + 4az3 - 15az5 + 8az7 - a2 - 2a2z2 + 12a2z4 - 21a2z6 + 8a2z8 + 3a3z - 5a3z3 + 12a3z5 - 18a3z7 + 6a3z9 + 2a4 - 19a4z2 + 47a4z4 - 33a4z6 + 2a4z8 + 2a4z10 + 5a5z - 29a5z3 + 59a5z5 - 42a5z7 + 9a5z9 + 3a6 - 19a6z2 + 31a6z4 - 10a6z6 - 5a6z8 + 2a6z10 + 2a7z - 16a7z3 + 27a7z5 - 16a7z7 + 3a7z9 + a8 - 5a8z2 + 8a8z4 - 5a8z6 + a8z8

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

{1, 0}

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=0 is the signature of 11313. 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 = 7                       1 j = 5                     2   j = 3                   3 1   j = 1                 5 2     j = -1               6 4       j = -3             6 4         j = -5           5 6           j = -7         5 6             j = -9       3 5               j = -11     2 5                 j = -13   1 3                   j = -15   2                     j = -17 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, 313]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 313]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[18, 11, 19, 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, 313]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -1, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -6, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 313]]
Out[5]=	DTCode[6, 12, 16, 14, 20, 18, 2, 22, 4, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 313]][t]
Out[6]=	5 19 2 29 + -- - -- - 19 t + 5 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 313]][z]
Out[7]=	2 4 1 + z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 313]}
In[9]:=	{KnotDet[Knot[11, Alternating, 313]], KnotSignature[Knot[11, Alternating, 313]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 313]][q]
Out[10]=	-8 3 5 8 10 11 12 10 2 3 8 + q - -- + -- - -- + -- - -- + -- - -- - 5 q + 3 q - q 7 6 5 4 3 2 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, 313]}
In[12]:=	A2Invariant[Knot[11, Alternating, 313]][q]
Out[12]=	-26 -24 2 -18 3 2 -10 2 2 -2 2 4 q + q - --- - q - --- + --- + q + -- + -- - q + 2 q - 2 q + 22 16 14 8 4 q q q q q 6 8 10 > q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 313]][a, z]
Out[13]=	2 2 4 6 8 z 2 2 4 2 6 2 4 2 4 a + 2 a - 3 a + a - -- + 2 a z + 3 a z - 3 a z + z + 2 a z + 2 a 4 4 > 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 313]][a, z]
Out[14]=	2 2 4 6 8 3 5 7 2 z 2 2 -a + 2 a + 3 a + a + 3 a z + 5 a z + 2 a z + 2 z - -- - 2 a z - 2 a 3 3 4 2 6 2 8 2 z 3 z 3 3 3 5 3 > 19 a z - 19 a z - 5 a z + -- - ---- + 4 a z - 5 a z - 29 a z - 3 a a 4 5 7 3 4 3 z 2 4 4 4 6 4 8 4 5 z > 16 a z - 9 z + ---- + 12 a z + 47 a z + 31 a z + 8 a z + ---- - 2 a a 5 3 5 5 5 7 5 6 2 6 4 6 > 15 a z + 12 a z + 59 a z + 27 a z + 7 z - 21 a z - 33 a z - 6 6 8 6 7 3 7 5 7 7 7 2 8 > 10 a z - 5 a z + 8 a z - 18 a z - 42 a z - 16 a z + 8 a z + 4 8 6 8 8 8 3 9 5 9 7 9 4 10 > 2 a z - 5 a z + a z + 6 a z + 9 a z + 3 a z + 2 a z + 6 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 313]], Vassiliev[3][Knot[11, Alternating, 313]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 313]][q, t]
Out[16]=	4 1 2 1 3 2 5 3 5 - + 5 q + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + q 17 8 15 7 13 7 13 6 11 6 11 5 9 5 9 4 q t q t q t q t q t q t q t q t 5 6 5 6 6 4 6 3 > ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + 3 q t + 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 7 3 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a313

K11a312 K11a314