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

K11a306 K11a308

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 16 20 18 4 2 22 10 8

Alexander Polynomial:

2t-3 - 9t-2 + 19t-1 - 23 + 19t - 9t2 + 2t3

Conway Polynomial:

1 + z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1083, K11a323, ...}

Determinant and Signature:

{83, -2}

Jones Polynomial:

q-9 - 3q-8 + 5q-7 - 8q-6 + 11q-5 - 13q-4 + 13q-3 - 11q-2 + 9q-1 - 5 + 3q - q2

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 2z2 - z4 + a2 + 3a2z2 + 3a2z4 + a2z6 + 2a4 + 4a4z2 + 3a4z4 + a4z6 - 3a6 - 5a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

- 2a-1z3 + a-1z5 + 3z2 - 7z4 + 3z6 + 3az3 - 8az5 + 4az7 - a2 - a2z2 + 6a2z4 - 8a2z6 + 4a2z8 + 2a3z - 5a3z3 + 8a3z5 - 6a3z7 + 3a3z9 + 2a4 - 17a4z2 + 32a4z4 - 19a4z6 + 4a4z8 + a4z10 + 2a5z - 15a5z3 + 29a5z5 - 20a5z7 + 6a5z9 + 3a6 - 18a6z2 + 33a6z4 - 22a6z6 + 4a6z8 + a6z10 - a7z + 2a7z3 + 2a7z5 - 7a7z7 + 3a7z9 + a8 - 4a8z2 + 11a8z4 - 13a8z6 + 4a8z8 - a9z + 7a9z3 - 10a9z5 + 3a9z7 + a10z2 - 3a10z4 + a10z6

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=-2 is the signature of 11307. 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                   3 1   j = -1                 6 2     j = -3               6 4       j = -5             7 5         j = -7           6 6           j = -9         5 7             j = -11       3 6               j = -13     2 5                 j = -15   1 3                   j = -17   2                     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, 307]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 307]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[4, 13, 5, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 307]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -6, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 307]]
Out[5]=	DTCode[6, 12, 14, 16, 20, 18, 4, 2, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 307]][t]
Out[6]=	2 9 19 2 3 -23 + -- - -- + -- + 19 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 307]][z]
Out[7]=	2 4 6 1 + z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 83], Knot[11, Alternating, 307], Knot[11, Alternating, 323]}
In[9]:=	{KnotDet[Knot[11, Alternating, 307]], KnotSignature[Knot[11, Alternating, 307]]}
Out[9]=	{83, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 307]][q]
Out[10]=	-9 3 5 8 11 13 13 11 9 2 -5 + 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, 307]}
In[12]:=	A2Invariant[Knot[11, Alternating, 307]][q]
Out[12]=	-28 -24 -22 2 -18 -16 2 2 -8 2 2 3 4 q - q + q - --- - q + q - --- + --- + q + -- - -- + -- + q - 20 14 12 6 4 2 q q q q q q 6 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 307]][a, z]
Out[13]=	2 4 6 8 2 2 2 4 2 6 2 8 2 4 a + 2 a - 3 a + a - 2 z + 3 a z + 4 a z - 5 a z + a z - z + 2 4 4 4 6 4 2 6 4 6 > 3 a z + 3 a z - 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 307]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 2 2 2 -a + 2 a + 3 a + a + 2 a z + 2 a z - a z - a z + 3 z - a z - 3 4 2 6 2 8 2 10 2 2 z 3 3 3 > 17 a z - 18 a z - 4 a z + a z - ---- + 3 a z - 5 a z - a 5 3 7 3 9 3 4 2 4 4 4 6 4 > 15 a z + 2 a z + 7 a z - 7 z + 6 a z + 32 a z + 33 a z + 5 8 4 10 4 z 5 3 5 5 5 7 5 > 11 a z - 3 a z + -- - 8 a z + 8 a z + 29 a z + 2 a z - a 9 5 6 2 6 4 6 6 6 8 6 10 6 > 10 a z + 3 z - 8 a z - 19 a z - 22 a z - 13 a z + a z + 7 3 7 5 7 7 7 9 7 2 8 4 8 > 4 a z - 6 a z - 20 a z - 7 a z + 3 a z + 4 a z + 4 a z + 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 4 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 307]], Vassiliev[3][Knot[11, Alternating, 307]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 307]][q, t]
Out[16]=	4 6 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 5 7 6 6 7 5 6 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 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a307

K11a306 K11a308