9 42

9_41

9_43

Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42's page at Knotilus! Visit 9 42's page at the original Knot Atlas!

9 42 Quick Notes

9_42 is Alexander Stoimenow's favourite knot!

Alsacian chair, alsacian museum, Strasbourg, France

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17
Gauss code	-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -18 -6 -12
Conway Notation	[22,3,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-5] Hyperbolic Volume 4.05686 A-Polynomial See Data:9 42/A-polynomial

[edit Notes for 9 42's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 42's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+2t-1+2t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 2 }
Jones polynomial	$q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

Further Quantum Invariants

Further quantum knot invariants for 9_42.

A1 Invariants.

Weight	Invariant
1	$q^{7}+q^{-7}$
2	$q^{22}-q^{18}+q^{6}+1+q^{-6}-q^{-18}+q^{-22}$
3	$q^{45}-q^{41}-q^{39}+q^{35}+q^{23}+q^{21}-q^{19}-q^{17}+q^{13}-q^{9}+q^{5}+q^{3}+q^{-3}+q^{-5}-q^{-13}-q^{-15}+q^{-21}+2q^{-23}-q^{-27}+2q^{-31}-3q^{-35}-q^{-37}+q^{-39}+q^{-41}$
4	$q^{76}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}-q^{58}+q^{48}+q^{46}-2q^{42}-2q^{40}+q^{36}+2q^{34}-q^{30}-q^{28}+2q^{24}+q^{22}+q^{20}-q^{18}-2q^{16}+q^{12}+2q^{10}-2q^{6}-q^{4}+2+q^{-2}-q^{-4}+q^{-8}+2q^{-10}+q^{-12}-q^{-14}+q^{-20}-q^{-24}-q^{-34}-2q^{-36}+q^{-40}+2q^{-42}+2q^{-44}-q^{-46}-q^{-48}-2q^{-50}+q^{-52}+3q^{-54}-2q^{-60}-q^{-62}+q^{-68}$
5	$q^{115}-q^{111}-q^{109}-q^{107}+q^{103}+2q^{101}+q^{99}-q^{95}-q^{93}-q^{91}+q^{87}+q^{81}+q^{79}-q^{75}-3q^{73}-2q^{71}+2q^{67}+3q^{65}+2q^{63}-2q^{59}-2q^{57}-q^{55}+q^{53}+2q^{51}+2q^{49}+q^{47}-q^{45}-2q^{43}-3q^{41}-2q^{39}+q^{37}+3q^{35}+3q^{33}+q^{31}-2q^{29}-4q^{27}-2q^{25}+q^{23}+3q^{21}+4q^{19}+q^{17}-2q^{15}-3q^{13}-q^{11}+2q^{9}+3q^{7}+2q^{5}-q^{3}-3q-2q^{-1}+q^{-3}+3q^{-5}+2q^{-7}-2q^{-11}-q^{-13}+q^{-15}+2q^{-17}+q^{-19}-q^{-21}-q^{-23}+q^{-27}-q^{-31}-q^{-33}+q^{-37}+q^{-39}+q^{-55}+2q^{-57}+q^{-59}-2q^{-61}-4q^{-63}-4q^{-65}+5q^{-69}+6q^{-71}+q^{-73}-5q^{-75}-6q^{-77}-3q^{-79}+3q^{-81}+6q^{-83}+5q^{-85}-3q^{-89}-3q^{-91}-2q^{-93}+q^{-97}+q^{-99}$

A2 Invariants.

Weight	Invariant
1,0	$q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}$
1,1	$q^{28}+2q^{24}+2q^{20}-2q^{18}-2q^{16}-2q^{14}-2q^{12}+2q^{6}+4q^{4}+4q^{2}+2+2q^{-2}-2q^{-4}-4q^{-8}-2q^{-12}+2q^{-14}+2q^{-18}+2q^{-24}-2q^{-26}+q^{-28}-2q^{-30}+2q^{-32}$
2,0	$q^{28}+q^{26}+q^{24}-q^{18}-q^{16}-2q^{14}-2q^{12}-q^{10}+q^{8}+3q^{6}+2q^{4}+3q^{2}+2+q^{-2}-q^{-4}-q^{-6}-q^{-8}-q^{-10}+q^{-16}+q^{-20}-q^{-22}-q^{-24}+q^{-28}+q^{-30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}+q^{16}+q^{14}+q^{12}-q^{4}-q^{-4}+q^{-12}+q^{-14}+q^{-16}+q^{-20}$
1,0,0	$q^{13}+q^{11}+2q^{9}+q^{7}-q^{3}-2q-2q^{-1}-q^{-3}+q^{-7}+2q^{-9}+q^{-11}+q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{26}+q^{24}+2q^{22}+2q^{20}+2q^{18}+q^{16}-2q^{12}-3q^{10}-3q^{8}-2q^{6}-q^{4}+q^{2}+4+4q^{-2}+3q^{-4}+2q^{-6}-3q^{-10}-3q^{-12}-3q^{-14}-2q^{-16}+2q^{-20}+3q^{-22}+2q^{-24}+2q^{-26}+q^{-28}-q^{-32}$
1,0,0,0	$q^{16}+q^{14}+2q^{12}+2q^{10}+q^{8}-q^{4}-2q^{2}-3-2q^{-2}-q^{-4}+q^{-8}+2q^{-10}+2q^{-12}+q^{-14}+q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}+q^{16}+q^{14}+q^{12}-q^{4}-2-q^{-4}+q^{-12}+q^{-14}+q^{-16}+q^{-20}$
1,0	$q^{34}+q^{26}+q^{18}-q^{14}+1-q^{-14}+q^{-18}+q^{-26}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{26}+q^{22}+q^{20}+2q^{18}+q^{16}+q^{14}+q^{12}-q^{6}-q^{4}-2q^{2}-2q^{-2}-q^{-4}-q^{-6}+q^{-12}+q^{-14}+q^{-16}+2q^{-18}+q^{-20}+q^{-22}+q^{-26}$

G2 Invariants.

Weight	Invariant
1,0	$q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}$

.

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["9 42"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 7, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$

Vassiliev invariants

V2 and V3:

(-2, 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 $-8$ $0$ $32$ ${\frac {164}{3}}$ ${\frac {76}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1312}{3}}$ $-{\frac {608}{3}}$ $-{\frac {6271}{15}}$ ${\frac {1484}{15}}$ $-{\frac {19564}{45}}$ ${\frac {607}{9}}$ $-{\frac {1471}{15}}$

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

\   r    \    j   \	-4	-3	-2	-1	0	1	2	χ
7							1	1
5								0
3					1	1		0
1				1	1			0
-1				1	1			0
-3		1	1					0
-5								0
-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[9, 42]]
Out[2]=	9
In[3]:=	PD[Knot[9, 42]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[18, 14, 1, 13], X[12, 18, 13, 17]]
In[4]:=	GaussCode[Knot[9, 42]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8]
In[5]:=	BR[Knot[9, 42]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 42]][t]
Out[6]=	-2 2 2 -1 - t + - + 2 t - t t
In[7]:=	Conway[Knot[9, 42]][z]
Out[7]=	2 4 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 42]}
In[9]:=	{KnotDet[Knot[9, 42]], KnotSignature[Knot[9, 42]]}
Out[9]=	{7, 2}
In[10]:=	J=Jones[Knot[9, 42]][q]
Out[10]=	-3 -2 1 2 3 -1 + q - q + - + q - q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 42]}
In[12]:=	A2Invariant[Knot[9, 42]][q]
Out[12]=	-10 -8 -6 -2 2 6 8 10 -1 + q + q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[9, 42]][a, z]
Out[13]=	2 3 2 2 2 z 2 6 z 2 2 6 z 3 -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + 6 a z - 2 a 2 a a a 4 5 6 7 4 5 z 2 4 5 z 5 6 z 2 6 z 7 10 z - ---- - 5 a z - ---- - 5 a z + 2 z + -- + a z + -- + a z 2 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 42]], Vassiliev[3][Knot[9, 42]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[9, 42]][q, t]
Out[15]=	1 3 1 1 1 1 q 3 7 2 - + q + q + ----- + ----- + ----- + --- + - + q t + q t q 7 4 3 3 3 2 q t t q t q t q t