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

K11a112 K11a114

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X8,17,9,18 X12,19,13,20 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 18 2 20 22 8 12 6

Alexander Polynomial:

t-4 - 6t-3 + 15t-2 - 21t-1 + 23 - 21t + 15t2 - 6t3 + t4

Conway Polynomial:

1 + z2 - z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{109, -4}

Jones Polynomial:

q-10 - 3q-9 + 6q-8 - 11q-7 + 15q-6 - 17q-5 + 17q-4 - 15q-3 + 12q-2 - 7q-1 + 4 - q

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

{...}

A2 (sl(3)) Invariant:

q-30 - q-26 + q-24 - 3q-22 + 2q-20 - q-18 - q-16 + 2q-14 - 4q-12 + 4q-10 - q-8 + 2q-6 + 2q-4 - q-2 + 2 - q2

HOMFLY-PT Polynomial:

a2 - a2z2 - 3a2z4 - a2z6 + 2a4 + 8a4z2 + 9a4z4 + 5a4z6 + a4z8 - 3a6 - 9a6z2 - 8a6z4 - 2a6z6 + a8 + 3a8z2 + a8z4

Kauffman Polynomial:

2az3 - 3az5 + az7 - a2 - 4a2z2 + 16a2z4 - 15a2z6 + 4a2z8 + a3z - 2a3z3 + 12a3z5 - 16a3z7 + 5a3z9 + 2a4 - 17a4z2 + 43a4z4 - 36a4z6 + 5a4z8 + 2a4z10 - 6a5z3 + 24a5z5 - 32a5z7 + 11a5z9 + 3a6 - 19a6z2 + 43a6z4 - 40a6z6 + 9a6z8 + 2a6z10 - a7z + 4a7z3 - 2a7z5 - 8a7z7 + 6a7z9 + a8 - 4a8z2 + 11a8z4 - 14a8z6 + 8a8z8 + a9z + 3a9z3 - 8a9z5 + 7a9z7 + a10z2 - 4a10z4 + 5a10z6 + a11z - 3a11z3 + 3a11z5 - a12z2 + a12z4

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=-4 is the signature of 11113. 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 = 3                       1 j = 1                     3   j = -1                   4 1   j = -3                 8 3     j = -5               8 5       j = -7             9 7         j = -9           8 8           j = -11         7 9             j = -13       4 8               j = -15     2 7                 j = -17   1 4                   j = -19   2                     j = -21 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, 113]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 113]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 113]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -10, 7, -3, 8, -4, 9, -5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 113]]
Out[5]=	DTCode[4, 10, 14, 16, 18, 2, 20, 22, 8, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 113]][t]
Out[6]=	-4 6 15 21 2 3 4 23 + t - -- + -- - -- - 21 t + 15 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 113]][z]
Out[7]=	2 4 6 8 1 + z - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 113]}
In[9]:=	{KnotDet[Knot[11, Alternating, 113]], KnotSignature[Knot[11, Alternating, 113]]}
Out[9]=	{109, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 113]][q]
Out[10]=	-10 3 6 11 15 17 17 15 12 7 4 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q 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, 113]}
In[12]:=	A2Invariant[Knot[11, Alternating, 113]][q]
Out[12]=	-30 -26 -24 3 2 -18 -16 2 4 4 -8 2 2 + q - q + q - --- + --- - q - q + --- - --- + --- - q + -- + 22 20 14 12 10 6 q q q q q q 2 -2 2 > -- - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 113]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 2 4 a + 2 a - 3 a + a - a z + 8 a z - 9 a z + 3 a z - 3 a z + 4 4 6 4 8 4 2 6 4 6 6 6 4 8 > 9 a z - 8 a z + a z - a z + 5 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 113]][a, z]
Out[14]=	2 4 6 8 3 7 9 11 2 2 4 2 -a + 2 a + 3 a + a + a z - a z + a z + a z - 4 a z - 17 a z - 6 2 8 2 10 2 12 2 3 3 3 5 3 > 19 a z - 4 a z + a z - a z + 2 a z - 2 a z - 6 a z + 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 4 a z + 3 a z - 3 a z + 16 a z + 43 a z + 43 a z + 11 a z - 10 4 12 4 5 3 5 5 5 7 5 9 5 > 4 a z + a z - 3 a z + 12 a z + 24 a z - 2 a z - 8 a z + 11 5 2 6 4 6 6 6 8 6 10 6 7 > 3 a z - 15 a z - 36 a z - 40 a z - 14 a z + 5 a z + a z - 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 16 a z - 32 a z - 8 a z + 7 a z + 4 a z + 5 a z + 9 a z + 8 8 3 9 5 9 7 9 4 10 6 10 > 8 a z + 5 a z + 11 a z + 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 113]], Vassiliev[3][Knot[11, Alternating, 113]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 113]][q, t]
Out[16]=	5 8 1 2 1 4 2 7 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 8 7 9 8 8 9 7 8 3 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 4 t t 2 3 3 > --- + -- + 3 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a113

K11a112 K11a114