10 147

10_146

10_148

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10 147 Quick Notes

10 147 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X5,14,6,15 X15,20,16,1 X12,7,13,8 X8,18,9,17 X19,7,20,6 X16,12,17,11 X18,13,19,14 X2,10,3,9
Gauss code	1, -10, 2, -1, -3, 7, 5, -6, 10, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4
Dowker-Thistlethwaite code	4 10 -14 12 2 16 18 -20 8 -6
Conway Notation	[211,3,21-]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-7] Hyperbolic Volume 9.41759 A-Polynomial See Data:10 147/A-polynomial

[edit Notes for 10 147's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for 10 147's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+7t-9+7t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$q^{5}-3q^{4}+4q^{3}-4q^{2}+5q-4+3q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}+a^{2}z^{2}-z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}+2az^{7}+4z^{7}a^{-1}+2z^{7}a^{-3}+a^{2}z^{6}-z^{6}a^{-2}+z^{6}a^{-4}-z^{6}-8az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}-4a^{2}z^{4}-2z^{4}a^{-2}-6z^{4}+8az^{3}+13z^{3}a^{-1}+8z^{3}a^{-3}+3z^{3}a^{-5}+4a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-6}+6z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$
The A2 invariant	$q^{10}+q^{4}-q^{2}+2q^{-6}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
The G2 invariant	$q^{46}-q^{44}+3q^{42}-4q^{40}+3q^{38}-q^{36}-3q^{34}+10q^{32}-11q^{30}+12q^{28}-6q^{26}-5q^{24}+12q^{22}-14q^{20}+11q^{18}-5q^{16}-4q^{14}+12q^{12}-9q^{10}+3q^{8}+5q^{6}-13q^{4}+14q^{2}-7-5q^{-2}+10q^{-4}-14q^{-6}+18q^{-8}-11q^{-10}+4q^{-12}+4q^{-14}-13q^{-16}+15q^{-18}-13q^{-20}+7q^{-22}-7q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+6q^{-34}-14q^{-36}+11q^{-38}-q^{-40}-7q^{-42}+13q^{-44}-15q^{-46}+11q^{-48}+q^{-50}-6q^{-52}+7q^{-54}-11q^{-56}+7q^{-58}-3q^{-62}+2q^{-64}-2q^{-66}+q^{-68}+q^{-70}+2q^{-72}-q^{-74}-q^{-78}-q^{-84}+q^{-86}$

Further Quantum Invariants

Further quantum knot invariants for 10_147.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+q^{3}-q+q^{-1}+q^{-3}+q^{-7}-2q^{-9}+q^{-11}$
2	$q^{22}-q^{20}-2q^{18}+3q^{16}+q^{14}-4q^{12}+2q^{10}+4q^{8}-3q^{6}-2q^{4}+3q^{2}-1-3q^{-2}+3q^{-4}+3q^{-6}-2q^{-8}+q^{-10}+4q^{-12}-q^{-14}-5q^{-16}+2q^{-18}+q^{-20}-4q^{-22}+2q^{-24}+2q^{-26}-q^{-28}$
3	$q^{45}-q^{43}-2q^{41}+4q^{37}+3q^{35}-5q^{33}-6q^{31}+3q^{29}+9q^{27}+2q^{25}-10q^{23}-8q^{21}+6q^{19}+12q^{17}+q^{15}-13q^{13}-7q^{11}+11q^{9}+13q^{7}-6q^{5}-15q^{3}+3q+14q^{-1}+q^{-3}-15q^{-5}-q^{-7}+14q^{-9}+3q^{-11}-11q^{-13}-3q^{-15}+10q^{-17}+7q^{-19}-5q^{-21}-11q^{-23}-q^{-25}+10q^{-27}+8q^{-29}-12q^{-31}-14q^{-33}+7q^{-35}+19q^{-37}-q^{-39}-15q^{-41}-3q^{-43}+9q^{-45}+6q^{-47}-6q^{-49}-4q^{-51}+q^{-53}+2q^{-55}+q^{-57}-q^{-59}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{4}-q^{2}+2q^{-6}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
1,1	$q^{28}-2q^{26}+6q^{24}-12q^{22}+19q^{20}-28q^{18}+36q^{16}-38q^{14}+33q^{12}-24q^{10}+12q^{8}+14q^{6}-31q^{4}+48q^{2}-60+62q^{-2}-69q^{-4}+56q^{-6}-42q^{-8}+32q^{-10}-3q^{-12}-6q^{-14}+28q^{-16}-38q^{-18}+38q^{-20}-40q^{-22}+24q^{-24}-18q^{-26}+11q^{-28}-2q^{-30}-2q^{-32}+2q^{-34}+2q^{-36}-2q^{-42}+q^{-44}$
2,0	$q^{28}-q^{24}-q^{22}+q^{20}+2q^{18}-q^{16}-q^{14}+q^{12}+2q^{10}+q^{8}-2q^{6}+q^{2}-1-2q^{-2}+q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+2q^{-14}+2q^{-16}+2q^{-18}-q^{-20}-5q^{-22}-q^{-24}-2q^{-28}-q^{-30}+2q^{-32}+3q^{-34}-q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{16}+q^{14}-2q^{12}+2q^{10}-q^{6}+3q^{4}-1+2q^{-2}-2q^{-6}+q^{-8}+q^{-12}-q^{-14}+2q^{-18}-2q^{-20}+q^{-22}+3q^{-24}-2q^{-26}+2q^{-30}-2q^{-32}-q^{-34}+q^{-36}$
1,0,0	$q^{13}+q^{9}+q^{5}-q^{3}-q^{-1}+q^{-7}+2q^{-9}+q^{-13}-q^{-15}-q^{-19}+q^{-21}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{26}+2q^{20}+q^{18}-2q^{16}+q^{12}-2q^{10}-2q^{8}+3q^{6}+4q^{4}-q^{2}+2+6q^{-2}-4q^{-6}-q^{-10}-5q^{-12}-q^{-14}+2q^{-16}+q^{-18}-q^{-20}+4q^{-22}+3q^{-24}-2q^{-26}+3q^{-30}-2q^{-34}+q^{-36}+q^{-38}-q^{-40}-2q^{-42}+q^{-46}$
1,0,0,0	$q^{16}+q^{12}+q^{10}+q^{6}-q^{4}-1-q^{-2}+q^{-8}+q^{-10}+2q^{-12}+q^{-16}-q^{-18}-q^{-24}+q^{-26}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+3q^{16}-3q^{14}+4q^{12}-4q^{10}+4q^{8}-3q^{6}+q^{4}-3+4q^{-2}-6q^{-4}+8q^{-6}-7q^{-8}+8q^{-10}-5q^{-12}+5q^{-14}-2q^{-16}+2q^{-20}-3q^{-22}+3q^{-24}-4q^{-26}+4q^{-28}-4q^{-30}+2q^{-32}-q^{-34}+q^{-36}$
1,0	$q^{34}-q^{30}-q^{28}+2q^{26}+2q^{24}-2q^{22}-3q^{20}+q^{18}+4q^{16}+q^{14}-4q^{12}-2q^{10}+4q^{8}+4q^{6}-q^{4}-4q^{2}+3q^{-2}+q^{-4}-2q^{-6}-2q^{-8}+2q^{-10}+2q^{-12}-q^{-14}-3q^{-16}+2q^{-18}+4q^{-20}-4q^{-24}-q^{-26}+3q^{-28}+2q^{-30}-3q^{-32}-3q^{-34}+2q^{-36}+4q^{-38}-3q^{-42}-q^{-44}+2q^{-46}+3q^{-48}-q^{-50}-2q^{-52}-q^{-54}+q^{-58}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}-q^{24}+2q^{22}-2q^{20}+4q^{18}-3q^{16}+3q^{14}-3q^{12}+4q^{10}-2q^{8}+q^{6}+3-3q^{-2}+4q^{-4}-5q^{-6}+5q^{-8}-6q^{-10}+6q^{-12}-6q^{-14}+5q^{-16}-4q^{-18}+3q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+3q^{-30}-2q^{-32}+4q^{-34}-3q^{-36}+2q^{-38}-3q^{-40}+3q^{-42}-2q^{-44}-q^{-48}+q^{-50}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+3q^{42}-4q^{40}+3q^{38}-q^{36}-3q^{34}+10q^{32}-11q^{30}+12q^{28}-6q^{26}-5q^{24}+12q^{22}-14q^{20}+11q^{18}-5q^{16}-4q^{14}+12q^{12}-9q^{10}+3q^{8}+5q^{6}-13q^{4}+14q^{2}-7-5q^{-2}+10q^{-4}-14q^{-6}+18q^{-8}-11q^{-10}+4q^{-12}+4q^{-14}-13q^{-16}+15q^{-18}-13q^{-20}+7q^{-22}-7q^{-26}+11q^{-28}-6q^{-30}+3q^{-32}+6q^{-34}-14q^{-36}+11q^{-38}-q^{-40}-7q^{-42}+13q^{-44}-15q^{-46}+11q^{-48}+q^{-50}-6q^{-52}+7q^{-54}-11q^{-56}+7q^{-58}-3q^{-62}+2q^{-64}-2q^{-66}+q^{-68}+q^{-70}+2q^{-72}-q^{-74}-q^{-78}-q^{-84}+q^{-86}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 147"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-2t^{2}+7t-9+7t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{4}-z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{5}-3q^{4}+4q^{3}-4q^{2}+5q-4+3q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{4}a^{-2}-z^{4}+a^{2}z^{2}-z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}+a^{2}+a^{-2}-1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-2}+z^{8}+2az^{7}+4z^{7}a^{-1}+2z^{7}a^{-3}+a^{2}z^{6}-z^{6}a^{-2}+z^{6}a^{-4}-z^{6}-8az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}-4a^{2}z^{4}-2z^{4}a^{-2}-6z^{4}+8az^{3}+13z^{3}a^{-1}+8z^{3}a^{-3}+3z^{3}a^{-5}+4a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-6}+6z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1$

Vassiliev invariants

V2 and V3:

(-1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $0$ $8$ ${\frac {82}{3}}$ ${\frac {62}{3}}$ $0$ $32$ $0$ $32$ $-{\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {248}{3}}$ $-{\frac {1951}{30}}$ ${\frac {1022}{15}}$ $-{\frac {7022}{45}}$ ${\frac {703}{18}}$ $-{\frac {1471}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-4	-3	-2	-1	0	1	2	3	4	χ
11									1	1
9								2		-2
7							2	1		1
5						2	2			0
3					3	2				1
1				2	3					1
-1			1	2						-1
-3		1	2							1
-5		1								-1
-7	1									1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 147]]
Out[2]=	10
In[3]:=	PD[Knot[10, 147]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], X[12, 7, 13, 8], X[8, 18, 9, 17], X[19, 7, 20, 6], X[16, 12, 17, 11], X[18, 13, 19, 14], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 147]]
Out[4]=	GaussCode[1, -10, 2, -1, -3, 7, 5, -6, 10, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4]
In[5]:=	BR[Knot[10, 147]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, -3, 2, -1, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 147]][t]
Out[6]=	2 7 2 -9 - -- + - + 7 t - 2 t 2 t t
In[7]:=	Conway[Knot[10, 147]][z]
Out[7]=	2 4 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 11], Knot[10, 147], Knot[11, NonAlternating, 122]}
In[9]:=	{KnotDet[Knot[10, 147]], KnotSignature[Knot[10, 147]]}
Out[9]=	{27, 2}
In[10]:=	J=Jones[Knot[10, 147]][q]
Out[10]=	-3 2 3 2 3 4 5 -4 + q - -- + - + 5 q - 4 q + 4 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 147]}
In[12]:=	A2Invariant[Knot[10, 147]][q]
Out[12]=	-10 -4 -2 6 10 12 14 16 q + q - q + 2 q + q - q - q + q
In[13]:=	Kauffman[Knot[10, 147]][a, z]
Out[13]=	2 2 -2 2 z 3 z 4 z 2 z z 2 2 -1 - a - a - -- - --- - --- - 2 a z + 6 z + -- + -- + 4 a z + 5 3 a 6 2 a a a a 3 3 3 4 5 5 3 z 8 z 13 z 3 4 2 z 2 4 6 z 14 z ---- + ---- + ----- + 8 a z - 6 z - ---- - 4 a z - ---- - ----- - 5 3 a 2 3 a a a a a 6 6 7 7 8 5 6 z z 2 6 2 z 4 z 7 8 z 8 a z - z + -- - -- + a z + ---- + ---- + 2 a z + z + -- 4 2 3 a 2 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 147]], Vassiliev[3][Knot[10, 147]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 147]][q, t]
Out[15]=	3 1 1 1 2 1 2 2 q 3 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 11 4 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t + q t